Saturday, October 30, 2021

The Correct Usage of Correlative Conjunctions

Correlative conjunctions are pairs of words that correlate two equally important clauses or phrases in one complete thought, that is, they link equivalent elements together (a verb to a verb, a noun to a noun, an adjective to an adjective).

Ang mga correlative conjunctions o mga pangatnig na nag-uugnay ay mga pares ng mga salita na nag-uugnay ng dalawang magkatulad na mahahalagang sugnay o parirala sa isang kumpletong kaisipan, ibig sabihin, pinag-uugnay nila ang mga katumbas na elemento (isang pandiwa sa isang pandiwa, isang pangngalan sa isang pangngalan, isang pang-uri sa pang-uri).

Correlative conjunctions are one of the three main types of conjunctions used in the English language to create smooth flow and reduce sentence fragments, along with coordinating conjunctions and subordinating conjunctions. Correlative conjunctions work in pairs to correlate two parts of a sentence of equal importance. Correlative conjunctions often connect two singular subjects with a singular verb, or two plural subjects with a plural verb. They apply a relation between two subjects or two verbs that act in tandem with each other.

 (https://www.masterclass.com/articles/correlative-conjunctions-explained#what-is-a-correlative-conjunction)

Examples of Correlative Conjunctions

AEither ... or - used in a sentence in the affirmative sense when referring to a choice between two possibilities. It connects two positive statements of equal weight and things of the same types, phrases, clauses, or words. It indicates that there is a choice between the two choices, and only one can be selected. The verb agrees with the noun that is closer to it.

Ang either/or (alinman/o)ay ginagamit sa isang pangungusap sa affirmative sense kapag tumutukoy sa isang pagpipilian sa pagitan ng dalawang posibilidad. Ito ay nag-uugnay ng dalawang positibong pahayag na may pantay na bigat at mga bagay na magkapareho ang uri, parirala, sugnay, o salita. Ito ay nagpapahiwatig na mayroong pipiliin sa pagitan ng dalawang pagpipilian, at isa lamang ang maaaring mapili. Ang pandiwa ay sumasang-ayon sa pangngalan na mas malapit dito.

Examples/Mga Halimbawa

1. Jenny likes to eat cake and ice cream but her money is just enough for one. What will she order?

        She will order either cake or ice cream.

2. Edward wants to follow one of his parents' vocations. His mother is a doctor and his father is a lawyer. Next year, he will be entering university. What will Edward study?

        Edward will study either medicine or law.

3. Half of the class likes to play basketball. The other half likes to play volleyball.

        The class is to play either basketball or volleyball.

4. The excursionists don’t have a lot of time today, so they can either visit the zoo or watch an exhibition.

B. Neither ... nor - is used in a sentence in the negative sense when you want to say that two or more things are not true. It is used to connect the same kind of word or phrase in a sentence. It connects two negative statements of equal weight about two people or things. The verb agrees with the noun that is closer to it. 

The combination neither/nor indicates that neither of the two choices can be selected. In other words, neither choice is available.

Ang neither / nor (wala/ ni  o hindi / ni) ay ginagamit sa isang pangungusap sa negatibong kahulugan kapag gusto mong sabihin na ang dalawa o higit pang mga bagay ay hindi totoo. Ito ay ginagamit upang ikonekta ang parehong uri ng salita o parirala sa isang pangungusap. Nag-uugnay ito ng dalawang negatibong pahayag na may pantay na timbang tungkol sa dalawang tao o bagay. Ang pandiwa ay sumasang-ayon sa simuno (subject) pangngalan man o panghalip (either noun or pronoun) na mas malapit dito.

Ang kumbinasyong wala/ni hindi/ni ay nagpapahiwatig na wala sa dalawang pagpipilian ang maaaring piliin. Sa madaling salita, walang available na pagpipilian.

Examples/Mga Halimbawa

1. Jenny does not want cake. She does not want ice cream, too.

        Jenny wants neither cake nor ice cream.

2. Edward does not want to follow any of his parents' vocations. His mother is a doctor and his father is a lawyer. Next year, he will be entering university. What will Edward study?

        Edward will study neither medicine nor law.

3. Half of the class does not like to play basketball. The other half does not like to play volleyball.

        The class plays neither basketball nor volleyball.

4. The excursionists have no free time today, so they can neither visit the zoo nor watch an exhibition.

C. Both ... and  = implies a correlation between two subjects that are performing the same action. It refers to two things or people together. The combination both/and indicates that the two items are equally presented and included. The grammar is both A and B, that is A and B represent both nouns, verbs, or adjectives. The verbs always take plural forms.

Ang both ... and (Pareho ...at) ay nagpapahiwatig ng ugnayan sa pagitan ng dalawang paksa na nagsasagawa ng parehong aksyon. Ito ay tumutukoy sa dalawang bagay o taong magkasama. Ang kumbinasyong pareho/at ay nagpapahiwatig na ang dalawang aytem ay pantay na ipinakita at kasama. Ang gramatika ay parehong A at B, iyon ay, ang A at B ay kumakatawan sa parehong mga pangngalan, pandiwa, o pang-uri. Ang mga pandiwa ay laging may maramihang anyo.

Examples/Mga Halimbawa

1. Thomas likes chocolate ice cream. He also likes strawberry ice cream.

    Thomas likes both chocolate and strawberry ice cream.

2. The excursionists have a lot of time today. They can visit the zoo. They can also watch the exhibition.

    The excursionists can both visit the zoo and watch the exhibition.

3. Anne plays the piano. Jessica plays the piano, too.

    Both Anne and Jessica play the piano.

4. After shopping, the twins are tired. They are also hungry.

    After shopping, the twins are both tired and hungry.

4. After shopping, the twins are tired. Their mother is also tired.

    After shopping, both the twins and their mother are tired. 

D. Not only...but also -  used to connect and emphasize 2 words or 2 phrases at the same position. Both two phrases are being presented by the writer as surprising or unexpected, with the second one being even more surprising than the first. We use “not only but also” to give more information.

Ang not only ... but also   (hindi lamang...kundi ... rin) ay ginagamit upang ikonekta at bigyang-diin ang 2 salita o 2 parirala sa parehong posisyon. Ang dalawang parehong parirala ay ipinakita ng manunulat bilang nakakagulat o hindi inaasahan, na ang pangalawa ay mas nakakagulat kaysa sa una. Ginagamit natin ang "hindi lamang... kundi ...rin" upang magbigay ng higit pang impormasyon.

It can be used to list adjective qualities, nouns or verbs, to show complementary qualities, quantities or actions, events, and states. We use it when we have two things and we want to give a little extra emphasis to the second thing because it’s even better, or even worse, or more surprising, or more impressive, or more shocking than the first thing.

The combination not only/but also is similar to both/and because it shows that both items presented are included. However, the item after not only is normally something we expect the speaker to say, whereas the item after but also is often something unexpected:

Ang kumbinasyong hindi lamang/kundi ay katulad din sa pareho/at dahil ipinapakita nito na ang parehong mga item na ipinakita ay kasama. Gayunpaman, ang item pagkatapos ng hindi lamang ay karaniwang isang bagay na inaasahan nating sasabihin ng tagapagsalita, samantalang ang item pagkatapos ng kundi ay kadalasang isang bagay na hindi inaasahan.

When using not only . . . but also in a sentence, parallelism should be the goal. It means that the words following both parts of this correlative conjunction (i.e., not only and but also) should belong to the same parts of speech. For example, if a verb follows not only, then a verb should also follow but also. The verb agrees with the noun that is closer to it.

Tandaan na gamitin ang pagkakasunod ng mga bahagi ng pangungusap kung gagamitin ang "not only ... but also" sa unahan o gitna ng pangungusap.

Examples/Mga Halimbawa

1. Beginning/Unahan ng Pangungusap

    a. Not only + verb + subjectbut also subject + verb.

     Not only are Donnalyn's children inquisitive, but also they are clever.

    b. Not only + verb + subjectbut + subject + also + verb

     Not only did Bong Bong cook dinuguan but he also served them.

     Not only did Guillermo Tolentino sculpt the UP Oblation but he also did the Bonifacio Monument.

    c. Not only + verb + subjectbut also

    Not only will they paint the inside of the villa but also the outside.

    d. Not only + subject + but also + subject + verb ...

    Not only George but also Terry has come to the seminar.

    Not only bananas but also avocado is rich in potassium.

2. Middle/Gitna ng Pangungusap

    a. Subject + Verb + not only + Adjective + but also + Adjective
        (Simuno + Pandiwa + not only + Pang-uri + but also + Pang-uri)

         Madelyn is not only beautiful but also intelligent.

    b. Subject + Verb + not only + Adverb + but also + Adverb
         (Simuno + Pandiwa + not only + Pang-abay + but also + Pang-abay)

        Donald dances not only gracefully but also effortlessly.

    cSubject Verb not only +  Noun + but also Noun
         (Simuno + Pandiwa + not only + Pangngalan + but also + Pangngalan)

        Sheila likes not only pancit but also spaghetti.

        Jenny likes to eat cake. She also likes to eat ice cream.
        Jenny likes to eat not only cake but also ice cream.

    d. Subject + not only + Verb + but also + Verb
         (Simuno +  not only + Pandiwa + but also + Pandiwa)

        Gabby not only writes music but also sings it.

Other Correlative Conjuctions

E. Whether/or = connects two possible actions of a subject. 

    Ang whether/or (kung ... o) ay nag-uugnay ng dalawang posibleng gawin ng simuno.

Examples/Mga Halimbawa

1. I was not sure whether Father would visit us or not.

2. I do not care whether Tommy comes to my birthday or not. 

3. Lucy may or may not come with us. We will have to go.

    We will have to go whether Lucy may come with us or not.

4. Danny does not like Faith. He will have to marry him.

    Donny will have to marry Faith whether he likes her or not.

    Whether or not he likes her, Donny will have to marry Faith.

F. Rather/than = presents a subject’s preference for one thing over another. 

    Ang rather/than (sa halip na) ay nagpapahayag ng pagpili ng simuno sa isang aksyon, tao, bagay, atbp. sa halip na ibang aksyon, tao, bagay, atbp.

Examples/ Mga Halimbawa

1. Cely would rather have tea than coffee.

2. Rene would rather call than text his mother.

3. My sister would rather cook than wash the dishes.

G. Such/that - connects two independent clauses in a way that applies a reason for an action. 

    Ang such.than [ganyan (ganito/ganoon)/kaya] ay nag-uugnay sa dalawang independiyenteng sugnay sa paraang naglalapat ng dahilan para sa isang aksyon.

Examples/Mga Halimbawa

1. It was a very hot afternoon. We stopped playing soccer.

    It was such a hot afternoon that we stopped playing soccer.

2. The lecture was very boring. The students felt asleep.

    The lecture was such boring that the students felt asleep.

3. Marie has very fine manners. Her classmates like her.

    Maris has such fine manners that her classmates like her.

H. As/as compares nouns using an adjective or an adverb.

    Ang as/as (kasing-) ay nagkukumpara ng panggalan gamit ang pang-uri o pang-abay.

Examples/Mga Halimbawa

1. The rose is fragrant. The sampaguita is fragrant.

    The rose is as fragrant as the sampaguita.

2. A lion can run 80 km/h. A cheetah can run 130 km/h.

    A lion cannot run as fast as a cheetah.

3. Norman weighs 60 kg. Ronnie weighs 60 kg.

    Norman is as heavy as Ronnie.

4. Sydney recorded 10oC yesterday. Melbourne recorded 8ooC yesterday.

    Sydney was not as cold as Melbourne yesterday.

I. Scarcely (Hardly)/when - used to combine or rewrite sentences denoting two simultaneous past actions.

    Ang scarcely (hardly)/when (Bahagya/nang) ay ginagamit upang pagsamahin o muling isulat ang mga pangungusap na nagsasaad ng dalawang magkasabay na nakaraang aksyon.

Examples/Mga Halimbawa

1. I reached the bus station. At once, the bus left.

    Scarcely/Hardly had I reached the bus station when the bus left.

2. As soon as Martha left the building, it collapsed.

    Scarcely/Hardly had Martha left the building when it collapsed.

3. No sooner did Tommy close the door than it rained heavily.

    Scarcely/Hardly had Tommy closed the door when it rained heavily.

J. Not/but is used when the subject has both the first and the second quality or the first quality is wrong, and the second is right.

Examples/Mga Halimbawa

1. Sheila does not have one mansion. She has two.

    Sheila owns not one but two mansions.

 2. Joemari cried loudly. It was not for sadness. It's for joy.

    Joemari cried loudly not for sadness but for joy

3.  I saw the dead bodies of a mother and her son.

    I saw not two dead bodies, but love and innocence.

5 Tips for Using Correlative Conjunctions 
(Source: https://www.masterclass.com)

There are a variety of helpful tips and rules for properly using these parts of speech in your sentences. Here are a few rules to follow when using correlative conjunctions.

1. Mind your subject-verb agreement. Subjects and verbs need to match when using correlative conjunctions. Singular subjects must match singular verbs, and plural subjects must match plural verbs. If you have multiple subjects, match the verb to the subject that is closest to the verb. For instance, youwould say: “Both the owner and his dogs run through the park,” instead of “Both the owner and his dogs runs through the park.”

Treat a subject that features either/or or neither/nor as singular if the elements after the conjunctions are singular. If one is plural, put it nearest to the verb and use a plural verb.

2. Ensure your pronoun agreement. Similar to subject-verb agreements, pronouns must also agree with their verbs when using correlative conjunctions. For example, “She plays tennis” is the correct agreement between pronoun and verb rather than “She play tennis.” This can be confusing when the pronoun’s antecedent is part of a correlative conjunction pair, however, if there is more than one subject, use the agreement for the closest noun or noun phrase. Let’s use the example sentence: “Not just my sister but my friends were all there as well.” In this case, “were” matches the plurality of “friends” rather than matching it to the singular “sister.”

3. Make sure your sentence has a parallel structure. Parallelism is important in grammatical structure for tracking the subjects of your sentences. Parallel structure deals with the grammatical form of your sentences, such as when discussing multiple items or making a list. For example, let’s look at the two sentences: “My mom not only likes to hike, but also is a fan of camping” and “My mom not only likes hiking but also camping.” In the second phrase, “hiking” and “camping” are parallel, while “to hike” and “camping” are not parallel with each other in the first sentence.

Position your correlative conjunctions in your sentence so the same type of word follows each one. In other words, use a parallel structure.

4. Use a comma with independent clauses. Only use a comma when your correlative conjunction separates two independent clauses, and avoid using it to separate the correlative conjunctions themselves. For example, let’s look at the two sentences: “Neither you, nor I should wear pastels,” and “Neither you nor I should wear pastels.” In this example, the latter is correct because both subjects depend on the same verb, which is “wear.”

Don't use a comma with a correlative conjunction unless the words after it could be a standalone sentence (i.e., contain a subject, a verb and convey a complete idea).

5. Watch out for double negatives. Neither/nor indicates a negative connotation, so be sure your main clause does not also contain a negative verb phrase on top of that. “I can’t neither see it nor hear it,” is not correct because “can’t” already provides a negative. The correct version would be: “I can neither see it nor hear it.”

Don't use a negative verb with neither/nor otherwise you'll create a double negative.


EXERCISES

Complete each sentence using the correct correlative conjunction pair from the parenthesis:

1. Shelly plans to take her annual leave  _________ in September _________ in December. (rather/than, whether / or, either / or, as / if)

2. _________ Mother is feeling happy _________ sad, she tries to keep a positive attitude. (Either / or, Whether / or, When / and, Neither/nor)

3. _________ had I taken my shoes off _________ I found out we had to leave again. (As / as, Rather / than, Scarcely/when, Whether / or)

4. _________ only is coconut water delicious, _________ it can be healthy. (Such/that, Whether / or, Not / but, Just as / so)

5. _________ I have roast beef for dinner, _____________________I cannot have ice cream for dessert. (Either/or, If /then, When / than, Whether / or)

6. _________ flowers _________ trees grow during warm weather. (Not only / or, Both / and, Not / but, Neither/nor)

7. _________ do we enjoy summer vacation, _________ we enjoy winter break. (Whether / or, Not only / but also, Either / or, As/as)

8. Terry is 5 feet tall. Her brother is also 5 feet tall. Terry is ______ tall _____ her brother. (either/or, neither/nor, both/an, as/as)

9. The clouds are very dark. It’s _________ going to snow _________ rain tonight. (neither/nor, as / if, either / or, as / as)

10. Savory dishes are _________ sweet _________ sour. (often / and, neither / nor, both / and, either/or)


Answers: 

Wednesday, October 27, 2021

Lesson 11 - Trigonometric Functions in Everyday Life: Introduction to Trigonometry in Taglish | ALS Module

LESSON 11 – TRIGONOMETRIC FUNCTIONS IN EVERYDAY LIFE

Sa mga nakaraang aralin , natutunan natin ang tungkol sa special at reference angles. Gayunman, dahil hindi lahat ng mga anggulo ay mga espesyal na anggulo o lahat sila ay may mga reference angles o anggulong sanggunian na 30 , 45 , o 60 °, natutunan naman natin ang paggamit ng isang scientific calculator  upang hanapin ang numeric values ng anim na trigonometric functions – sine, cosine, tangent, cosecant, secant, at cotangent. Sa ngayon, handa na tayong gamitin ang lahat ng ating natutunan upang tugunan ang mga problema  sa pang-araw-araw na buhay na may kinalaman sa trigonometric functions.



PAG-ARALAN AT SURIIN NATIN


Interesado si Kevin na sukatin ang taas ng puno sa larawan sa itaas. Paano niya ito nalutas? Tingnan natin kung paano niya ito ginawa.

Una, sinukat niya ang taas mula sa lupa hanggang sa antas ng kanyang mata. Ito ay 5 ft..

Pagkatapos ay gumamit siya ng isang protractor upang sukatin ang anggulo sa pagitan ng pahalang sa kanyang linya ng paningin at sa tuktok ng puno.  Ito ay may sukat nga 40o.  

Pagkatapos  noon ay sinukat niya ang kanyang distansya mula sa puno. Ito ay 12 ft.

Matapos ito, ginawa niya ang mga sumusunod na pagkalkula:

        tan 40o    opposite side / adjacent side  
                opposite side  / distance from the tree
                  opposite side  /  12 ft.
        opposite side = tan 40o (12 ft.)
             = 0.8391 (12 ft.)
             = 10.0692 or 10.07 ft.

Samakatuwid, ang taas ng puno ay 10.07 ft. + 5 ft. = 15.07 ft.  o 15 feet.

Muli nating subukin na sumagot ng isa pang problema upang matasa ang ating nalalaman.


Si Jack ay nasa tuktok ng isang tore na may taas na 100 talampakan. Tumingin siya paibaba sa  kalsada at nakita niya ang kanyang kotse na naka-parada di kalayuan sa tore. Gaano kalayo ang kanyang kotse sa pinakaibaba ng tore?

Lutasin natin ang problema ng step by step:

STEP 1 Determine the relationship of the given side with the given angle.                             (Tukuyin ang kaugnayan ng ibinigay na panig sa ibinigay na anggulo.)

In the right triangle in the picture, the height of the tower (100 ft.) is the side opposite the 60° angle. (Sa nabuong right triangle sa larawan, ang taas ng tore [100 ft] ay ang gilid katapat (opposite side) ng anggulong 60 degrees.)

STEP 2 Determine the relationship of the unknown side with the given angle.                     (Tukuyin ang kaugnayan ng hinahanap na gilid sa ibinigay na anggulo.)

The unknown side is the side adjacent to the 60° angle.
(Ang hinahanap na gilid ay ang gilid kalapit (adjacent) ng anggulong 60 degrees.)

STEP 3 Determine the appropriate trigonometric function to be used. Here we                     are going to use the cotangent function.
        (Tukuyin ang naaangkop na trigonometric function na gagamitin. Dito                     ay gagamitin natin ang cotangent function.)

        cot 60° = adjacent side / opposite side

STEP 4 Substitute the given values and solve for the unknown. (Ihalili ang                             ibinigay na mga value at hanapin ang nawawalang gilid.)

        cot 60°  =  adjacent side /  100 ft
        adjacent side = cot 60° (100 ft)

(Gamit ang iyong scientific calculator, ating makukuha na ang numeric value ng cot 60o = 0.5774 [rounded-off to 4 decimal places]).

        adjacent side = 0.5774 (100 ft)
        adjacent side = 57.74 ft

Samakatuwid, ang kotse ni Jack ay 57.74 talampakan ang layo mula sa base ng tore.

Paano kung sa halip na cotangent ay tangent ang ating ginamit  na trigonometric function? Tingnan natin ang pagkalkula.

        tan 60° = opposite  side / adjacent side
        tan 60° =  100 ft /  adjacent side
        adjacent side (tan 60°) = 100 ft
        adjacent side = 100 ft / tan 60°

(Gamit ang iyong scientific calculator, ating makukuha na ang numeric value ng tan 60o = 1.73205080757.)

        adjacent side = 100 ft / 1.73205080757
        adjacent side = 57.74 ft (rounded-off to 2 decimal places)

Samakatuwid, kahit tangent ang ginamit sa pagkalkula, ang kotse ni Jack ay 57.74 talampakan pa rin ang layo mula sa base ng tore.

SUBUKIN NATIN ITO

Upang matasa pa ang ating natutunan, subukin nating sumagot ng isa pang problema. 


Suppose a man is standing on top of a 35 ft. building. He looks down to an open manhole and estimates that the angle from the horizontal down to the manhole is 63°. If the man is 5 ft. tall, how far is the manhole from the base of the building?

STEP 1 Determine the relationship of the given side with the given angle.

        In the right triangle represented in the picture above, the height of the                 building, 35 ft., is the side opposite the 63°angle.

STEP 2 Determine the relationship of the unknown side with the given angle.

        The unknown side is the side adjacent to the 63° angle.

STEP 3 Determine the appropriate trigonometric function to be used.
        tan 63° = opposite side / adjacent side

STEP 4 Substitute the given values and solve for the unknown.

        tan 63° = opposite side / adjacent side
        tan 63° = (height of building + height of man) / adjacent side
        tan 63° = (35 ft + 5 ft) / adjacent  side
        tan 63° = 40 ft / adjacent side
(Using a scientific calculator, the value of tan 63° = 1.9626)

        1.9626 =  40 ft / adjacent side
        adjacent  side ( 1.9626)  =  40 ft 
        adjacent side = 40 ft / 1.9626
        adjacent = 20.38 ft (rounded-off to 2 decimal places)

      Thus, the manhole is 20.38 feet from the base of the building.

SUBUKIN NATIN IYO

You can also solve problems involving trigonometric functions by using a shortcut method. (Masasagot din natin ang mga problemang kinapapalooban ng trigonometric functions sa pamamagitan ng shortcut method.)

A scuba diver makes an angle of 50° with the vertical when diving into an ocean. How far must he swim to be 100 meters below the water surface?


A right triangle is formed with the acute angle 50°. The distance from the water surface to the desired position of the diver (100 meters) is the side adjacent to 50°. We are asked for the length of the hypotenuse.

We will use the secant function. Recall that secant is the reciprocal/inverse of cosine. Cosine involves adjacent side and hypotenuse. Remember SohCahToa?

        sec 50° = hypotenuse / adjacent side
        sec 50° = distance the diver must swim / 100 m
        distance the diver must swim = sec 50o (100 m)

(Using a scientific calculator, sec 50° = 1.5557)

                distance the diver must swim = 1.5557 (100 m)

     Thus, the distance the diver must swim is 155.57 m  or 155.6 meters.

SUMMARY

A. We do the following steps in solving problems involving right triangles:

STEP 1 Determine the relationship of the given side with the given angle.
STEP 2 Determine the relationship of the unknown side with the given angle. 
STEP 3 Determine the appropriate trigonometric function to be used.
STEP 4 Substitute the given values and solve for the unknown.

B. Upang madaling matandaan ang kaugnayan ng isang ibinigay na angle sa hinahanap o ibinigay na gilid o side, tandaan ito: SohCahToa, kung saan ==>

S = sine o = opposite side
C = cosine a = adjacent side
T = tangent h = hypotenuse

sine         = opposite side / hypotenuse
cosine     = adjacent side  / hypotenuse
tangent   = opposite side / adjacent side

C. Tandaan ang reciprocal/inverse  o kabaliktaran ng sine, cosine, at tangent.

1. Ang kabaliktaran ng sine ay cosecant. Kung kaya,

    sine         =     opposite side / hypotenuse
    cosecant =     hypotenuse / opposite side

2. Ang kabaliktaran ng cosine ay secant, Kung kaya,

    cosine     =     adjacent side  / hypotenuse
    secant     =     hypotenuse / adjacent side

3. Ang kabaliktaran ng tangent ay cotangent. Kung kaya,  

    tangent     =     opposite side / adjacent side
    cotangent =     adjacent side / opposite side

D. Pareho rin ang makukuhang sagot kung cosecant sa halip na sine, secant sa halip na cosine, o cotangent sa halip na tangent ang ginamit na trigonometric function hangga’t tama ang pagkaka-set up ng equation. Sa parteng ito, kakailanganin ninyo ang natutunan sa Algebra. Dapat ay alam mag-cross multiplication at pagta-transfer ng mga variables from left to right and vice versa.

E. Upang maiwasan ang pagkakamali sa pagpili ng trigonometric function base sa given at unknown side, gamitin ang nakasulat sa table:


Nais ipahiwatig ng nasa itaas na kung ang NUMERATOR ng function ang hinahanap, iyon ang gamiting trigionometric function. Halimbawa, kung ang given side ay isang hypotenuse at ang unknown ay adjacent side, cosine ang gagamitin pero kung ang unknown ay opposite side at ang given ay hypotenuse, sine ang gagamitin.

F. Kung hindi ma-visualize ang isang problema, iguhit muna ito at lagyan ng label upang matiyak na tama ang iyong pagkakaunawa sa problema.

G. Sa ngayon, ang araling ito ay nakapokus lamang sa right triangle.

H. Alamin ang rules sa rounding-off para hindi magkamali sa pinal na sagot  kung ito ang hinihingi ng problema. 

PAGSASANAY

Use trigonometric functions to solve the following problems.

1. A surveyor wants to find the height of a tree. He measures the angle between his line of sight and the top of the tree and finds it to be 47°. The man is 5 feet (ft.) tall and he is 10 meters (m) away from the tree. What is the height of the tree?

2. From a lighthouse 35 m above sea level, the angle from the horizontal to the ship is 25°. How far is the boat from the top of the lighthouse?


3. Mike wants to find the height of a tree. He measures the angle from the horizontal to the top of the tree and finds it to be 47°. He is 1 m tall and 10 meters away from the tree. What is the height of the tree?

4. A scuba diver makes an angle of 46° with the vertical when diving into an ocean. How deep is he in the water if he swims 110 m?

5. A ladder is leaning against a wall at an angle of 54° with the ground. If the top of the ladder is 2 m from the ground, how long is the ladder?

6. A family picture is hung on the wall. Jack noticed that the angle of the picture from the horizontal is 25°. He is 1 m tall. If he is 3 m away from the wall, how high is the picture?

7. Mark is flying a kite, lies down on the ground and realizes that 300 feet of string are out. The angle of the string with the ground is 42.5°. How high is Mark's kite above the ground?

8. A 20 foot ladder rests against a wall. The ladder makes a 55° angle with the ground. How far from the wall is the base of the ladder?

9. The mast  is 9 meters high and a wire is stretched tight to form a straight line to the top of the mast at an  angle of 60°. How long is the wire in meters? (Source: https://brilliant.org)

10. A kite is flying at a height of 65 m attached to a string that is fixed at the base of a tree. If the inclination of the string with the ground is 31°, how far is the kite from the tree?

ANSWERS

Sunday, October 24, 2021

Lesson 10 - Using a Scientific Calculator: Sin, Cos, and Tan Keys: Undertanding Trigonometry in Taglish

LESSON 10 – USING A SCIENTIFIC CALCULATOR: SIN, COS AND TAN KEYS

Sa Lesson 8 at 9 , natutunan natin ang tungkol sa special at reference angles. Gayunman, hindi lahat ng mga anggulo ay mga espesyal na anggulo o lahat sila ay may mga reference angles o anggulong sanggunian na 30 °, 45 , o 60 °. Sa katunayan, karamihan sa mga anggulo  ay hindi espesyal. Samakatuwid, kinakailangang malaman kung paano makuha ang mga values ng mga trigonometric function ng mga anggulong ito gamit ang isang pang-agham o scientific  calculator.


Sa araling ito, ituturo sa iyo kung paano matukoy ang mga values ng mga trigonometric functions (sine, cosine, tangent, cosecant, secant, at cotagent) ng anumang naibigay na anggulo gamit ang iyong pang-agham na calculator. 


Ginagamit ang mga scientific calculators sa mga pagkalkula na kinasasangkutan ng mga trigonometric functions, pati na rin sa iba pang mga aplikasyon  sa istatistika at calculus.

PAG-ARALAN AT SURIIN NATIN


Ang larawan sa itaas ay isa lamang sa mga uri ng scientific calculator na mabibili sa merkado. Pansinin na taliwas sa ordinaryong calculator, ang mga pang-agham na calculator ay may karagdagang mga key. Ang mga ito  ay nagsasagawa ng maraming mga tungkulin na sinasaad ng

o
 depende sa gamit mong calculator. Ginagamit ang command o utos  na ito kapag may alternatibo o kahalili ang isang key function

Sa araling ito, bukod sa mga power at digit key, kakailanganin natin ang ilan sa mga karagdagang key na ito. Ito ay ang mga:

key sa ilang calculator.

Pagmasdan ang larawan sa ibaba.


Ayon sa larawan, ang anggulo na nagawa ng kotse sa tuktok ng gusali ay isang hindi espesyal na anggulo na sumusukat ng 37°. Paano natin malulutas ang value ng sin 37° gamit ang pang-agham na calculator?

Panahon na upang gamitin ang iyong scientific calculator. Tanungin ang iyong Instructional Manager o  guro na suriin kung tama ang iyong ginagawa.


SUBUKIN NATIN ITO

Upang matasa ang ating natutunan, subukin nating sumagot ng ilang problema hinggil sa paghanap ng value ng ilang trigonometric functions gamit ang iyong scientific calculator.


Handa ka na ngayon na sagutan ang pagsasanay sa ibaba.

PAGSASANAY A

Using your scientific calculator, find the value of the following trigonometric functions and round-off them to 4 decimal places:

1. sin 285°
2. sin 124°
3. cos 133°
4. tan 54°
5. tan 98°

PAG-ARALAN AT SURIIN NATIN

Muli nating pagmasdan ang larawan sa ibaba:




Sa nakaraan, hiniling sa atin na tukoyin ang value ng sine ng anggulo na nagawa ng kotse sa tuktok ng gusali. Nakakuha tayo ng sin 37° = 0.6018. Ano ang ibig sabihin ng value  na ito?

Mula sa  Lesson 7, natutunan natin na ang sine ng isang anggulo ay ang ratio ng gilid sa tapat  (opposite side) ng anggulong iyon sa hypotenuse  o

sin θ = opposite side/hypotenuse

Ito ay nangangahulugan na ang value ng sine sa sitwasyong ito ay ang ratio ng taas ng gusali (opposite side) sa distansya ng kotse mula sa tuktok ng gusali (hypotenuse).

Ngayon, ipagpalagay na nais nating makuha ang ratio ng distansya ng kotse mula sa tuktok ng gusali (hypotenuse) hanggang sa taas ng gusali (opposite side). Sa kasong ito, kakailanganin nating gamitin ang kabaligtaran o inverse ng sine function, na walang iba kundi ang cosecant.

csc θ = hypotenuse/opposite side

Kaya ngayon, handa na tayong hanapin ang value na ito sa pamamagitan ng paggamit ng function csc 37°.

Ngunit tulad ng nakikita mo, walang mga key na ipinahiwatig para sa csc at iba pang mga kabaligtaran o inverse functions. Dito natin gagamitin ang 
 key.

Paano gumagana ang key na ito? Kung pipindot 
tayo ng isang value sa calculator ang resulta ay magiging 1 divided by ng value na ating pinindot. Subukin natin ang sumusunod at tingnan mismo ang mangyayari.

Pindutin ang mga ito: 


Dapat nating makuha ang 0.25 , na katumbas ng 1 divided by the given value 4.

Alalahanin na ang csc θ = 1/sin θ

Upang makuha ang value ng csc 37°, pipindutin muna ang value ng sin 37° at pagkatapos ay pindutin ang inverse key. 




SUMMARY

1. To determine the value of the sine, cosine or tangent function of any angle, press the following keys on the scientific calculator:

1st: the measurement of given angle
2nd: the given trigonometric function (sin, cos or tan)

2.  To determine the value of the cosecant, secant or cotangent function of any angle, press the following keys on the scientific calculator:

1st: the measurement of the given angle
2nd: the reciprocal of the given trigonometric function 
3rd: the key

3.  If 1/x or x–1 is placed above the key, first press


4.  Scientific calculators differ. In some models, you have to press first the trigonometric function key before you input the measurement of the angle. The answer can be obtained by pressing either


5.  It is advised that you read the manual of your calculator and be familiar with its functions. At any rate, the values of the trigonometric functions being asked should be the same regardless of the brand, model, or make of your scientific calculator.

MGA SAGOT

Wednesday, October 20, 2021

Lesson 9 - Reference Angles: Introduction to Trigonometry in Taglish - ALS Module Trigonometric Functions 2

 LESSON 9 – REFERENCE ANGLES

Sa Lesson 9, matutunan natin ang tungkol sa reference angles o mga sangguniang anggulo. Ipapakita sa iyo ng araling ito kung paano hanapin ang mga values ng anim na trigonometric functions (sine, cosine, tangent, cosecant, secant, at cotangent) ng iba pang mga anggulo na may mga sukat na multiples ng 30o, 45o, at 60o. Malalaman mo na ang kanilang mga values ay katulad din ng natutunan natin sa nakaraang aralin.



PAG-ARALAN AT SURIIN NATIN

Pagmasdan ang Figure 1 sa ibaba:


Look at the terminal side of A (Angle A). What is the acute angle formed by this terminal side with the horizontal axis?

If your answer is B, you are right. B is the acute angle formed by the terminal side of ∠A and the horizontal axis. B is called the reference angle of ∠A. 

Pagmasdan ang Figure 2 sa ibaba:

Figure 2
(Image from analyzemath.com)

        Ang mga given angles A sa itaas ay may kulay itim, samantalang kulay pula naman ang mga reference angles.

Mapupuna natin ang mga reference angles ay mga acute angles (anggulo na may sukat na mataas sa zero degrees pero mas mababa sa 90 degrees.) Ang reference angle ay nabuo ng terminal side (pinakadulong gilid) ng given angle at ng horizontal axis ( or x-axis).

TANDAAN ANG MGA SUMUSUNOD

1. Lahat ng reference angle r ay acute angles na may sukat na mas mataas sa zero degree pero mas mababa sa 90 degrees.

2. Sa  First Quadrant (0o < θ < 90o), ang reference angle r ng given angle θ ay ang mismong given angle θ.

Halimbawa, kung ang given angle θ = 30o, ang reference angle r nito ay ∠30o din.

3. Sa Second Quadrant (90o < θ < 180o), mahahanap ang reference angle r ng given angle θ sa pamamagitan ng pagkuha ng supplementary angle nito o paggamit ng formula nito: r = 180o - θ
Halimbawa, kung θ = 130o , its reference angle r is:
r =  180o - θ ==> r = 180o – 130o ==> r =50o.

3. Sa Third Quadrant (180o < θ < 270o), mahahanap ang reference angle r ng given angle θ sa pamamagitan ng formulang ito:  r = θ - 180o

Halimbawa, kung θ = 250o , its reference angle r is:
r =  θ - 180==>   r = 250o – 180o ==> r =70o.

4. Sa Fourth Quadrant ( 270o < θ < 360o), mahahanap ang reference angle r ng given angle θ sa pamamagitan ng formulang ito:  r = 360o - θ

Halimbawa, kung θ = 305o , its reference angle r is:
r =  360o - θ==> r = 360o – 305o ==> r =55o.

SUBUKIN NATIN ITO

Upang matasa ang ating natutunan, subukin nating sumagot ng ilang problema hinggil sa paghanap ng reference angle r.

1. What is the reference angle of 120o?

STEP 1: Determine in which interval the given angle belongs.

120o is between 90o and 180o. Kung gayon, ang 120o ay matatagpuan sa Second Quadrant o Quadrant II.

STEP 2: Determine which formula for r will be used.

Dahil ang 120o ay nasa Second Quadrant, ang formula nating gagamitin ay 
r = 180o - θ

STEP 3: Substitute the value of the given angle.

ɵ = 120o
r = 180o - θ
r = 180o – 120o
r = 60o

Samakatuwid, ang reference angle r ng 120o ay ang special angle na 60o.


2. What is the reference angle of 330o?

STEP 1: Determine in which interval the given angle belongs.

300o is between 270o and 360o. Kung gayon ang 300o ay matatagpuan sa Fourth Quadrant o Quadrant IV.

STEP 2: Determine which formula for r will be used.

Dahil ang 330o ay nasa Fourth Quadrant, ang formula nating gagamitin ay 
r = 360o - θ

STEP 3: Substitute the value of the given angle.

ɵ = 330o
r = 360o - θ
r = 360o – 330o
r = 30o

Samakatuwid, ang reference angle r ng 330o ay ang special angle na 30o.

3. What is the reference angle of 240o?

STEP 1: Determine in which interval the given angle belongs.

240o is between 180o and 270o. Kung gayon ang 240o ay matatagpuan sa Third Quadrant o Quadrant III.

STEP 2: Determine which formula for r will be used.

Dahil ang 240o ay nasa Third Quadrant, ang formula nating gagamitin ay 
r = θ - 180o

STEP 3: Substitute the value of the given angle.

θ = 240o
        r = θ - 180o
r = 240o – 180o
r = 60o

Samakatuwid, ang reference angle r ng 240o ay ang special angle na 60o.

MATUTO TAYO

A. The numeric value of the trigonometric function of a given angle is equal to the numeric value of the trigonometric function of its reference angle.

Ibig sabihin, kung ang reference angle r ng Ao  ay Bo at  ang sin Ao = x, ang sin Bo ay x din.

When we say numeric value, it means that we also consider the sign (positive or negative) of the trigonometric function.

B. We note that a trigonometric function is positive or negative depending on the measure of the angle. In determining the sign of a numeric value, we observe the following rules:

    1. If the measure of θ is greater than 0° but less than 90°, all the six trigonometric functions (sin θ, cos θ, tan θ,  csc θ, sec θ , and cot θ)  are positive

Kung ang given angle θ ay nasa First Quadrant, lahat ng anim na trigonometric functions - sin θ, cos θ , tanθ,  csc θ, sec θ , at cot θ  - ay pawang  POSITIVE. 

    2. If the measure of θ is greater than 90° but less than 180°, sin θ and csc θ are positive. The other four functions are negative.

Kung ang given angle θ ay nasa Second Quadrant, ang sin θ  at ang kanyang reciprocal/inverse na csc θ ay POSITIVE; ang iba pang function ay NEGATIVE na.

    3. If the measure of θ is greater than 180° but less than 270°, tan θ and cot θ are positive. The other four functions are negative.

Kung ang given angle θ ay nasa Third Quadrant, ang tan θ at ang kanyang reciprocal / inverse na cot θ ay POSITIVE; ang iba pang function ay NEGATIVE na.

    4. If the measure of θ is greater than 270° but less than 360°, cos θ and sec θ are positive. The other four functions are negative.

Kung ang given angle θ ay nasa Fourth Quadrant, ang cos θ at ang kanyang reciprocal / inverse  na sec θ ay POSITIVE; ang iba pang function ay NEGATIVE na.

SUBUKIN NATIN ITO

Now let’s try to solve for the numeric values of the trigonometric functions of a given angle.

Problem 1: Find the values of the six trigonometric functions of the angle 120°. 

We have previously learned that the reference angle of 120° is 60°.
Therefore, we will use the trigonometric function values of the special angle 60° that you learned in Lesson 8. Since 120° is in the Second Quadrant, only sine and its reciprocal/inverse cosecant are positive.

Thus, we have:

sin 120° = sin 60° = √3/2 
csc 120° = csc 60° = 2/√3  𝑜𝑟 2√3/3
cos 120° = –cos 60° = - 1/2
sec 120° = –sec 60° = - 2/1  or -2
tan 120° = –tan 60° = - √3/1  𝑜𝑟 −√3
cot 120° = –cot 60°- 1/√3 𝑜𝑟 −√3/3

Problem 2: Find the values of the six trigonometric functions of the angle 225°

STEP 1: Determine the reference angle of the given angle.

225° is between 180° and 270° and in the Third Quadrant,  so we use the formula:
r = θ - 180° 
r = 225° - 180° 
r = 45°

Samakatuwid, ang reference angle ng 225o ay ang special angle na 45o.

STEP 2: Determine the values of the trigonometric functions of the reference angle. 

Gagamitin natin ang anim na trigonometric functions ng 45° na nakuha natin sa Lesson 8.  

STEP 3: Determine the signs of the values.

The given angle is 225° and in the Third Quadrant. Therefore, tangent and cotangent are the only positive functions. We thus have:

sin 225° = -sin 45° = - 1/√2 or - √2/2    
csc 225° = -csc 45° = - √2/1  𝑜𝑟 −√2
cos 225° = –cos 45° = - 1/√2  𝑜𝑟 −√2/2
sec 225°–sec 45°– √2/1  𝑜𝑟 −√2  
tan 225° = tan 45° = 1/1 or 1       
cot 225° = cot 45° = 1/1 or 1

Problem 3: Find the values of the six trigonometric functions of the angle 330°

STEP 1: Determine the reference angle of the given angle.

330° is between 270° and 360° and in the Fourth Quadrant,  so we use the formula:
r = 360o - θ 
r = 360° - 330° 
r = 30°

Samakatuwid, ang reference angle ng 330o ay ang special angle na 30o.

STEP 2: Determine the values of the trigonometric functions of the reference angle. 

Gagamitin natin ang anim na trigonometric functions ng 30° na nakuha natin sa Lesson 8.  

STEP 3: Determine the signs of the values.

The given angle is 330° and in the Fourth Quadrant. Therefore, cosine and secant are the only positive functions. We thus have:

sin 330° = -sin 30° = - 1/2
csc 330° = -csc 30° = - 2/1  𝑜𝑟 −2
cos 330° = cos 30° = √3/2
sec 330°sec 30°2/√3  𝑜𝑟  (2√3)/3
tan 330° = -tan 30° = - 1/√3 or - √3/3    
cot 330° = -cot 30° = - √3/1 or - √3

SUMMARY

Muling unawain at tandaan ang ating mga natutunan:

A. We use the following steps in computing for the reference angle r of a given angle θ:

STEP 1 Determine in which interval the given angle θ belongs or what quadrant                 it is located.

STEP 2 Determine which formula for r will be used.

STEP 3 Substitute the value of the given angle.

B. We use the corresponding formulas in computing for the reference angle according to the following rules.

    1. If the given angle θ is greater than 90° but less than 180° (90° < θ < 180°), the formula to be used is: 
r = 180°– θ .

    2. If the angle θ is greater than 180° but less than 270° (180° < θ < 270°), the formula to be used is: 
r = θ – 180°.

    3. If the given angle θ is greater than 270° but less than 360° (270° < θ < 360°), the formula to be used is: 
r = 360° – θ .

C. The numeric values of the trigonometric functions of a given angle is equal to the numeric values of the trigonometric functions of its reference angle.

In determining the signs of the numeric values of the trigonometric functions of a certain angle θ, we follow these rules:

    1. If the measure of θ is greater than 90° but less than 180°, sin θ and csc θ are positive. The other four functions are negative. The given angle θ is in the Second Quadrant.

    2. If the measure of θ is greater than 180° but less than 270°, tan θ and cot θ are positive. The other four functions are negative. The given angle θ is in the Third  Quadrant.

3. If the measure of θ is greater than 270° but less than 360°, cos θ and sec θ are positive. The other four functions are negative. The given angle θ is in the Fourth Quadrant.

4. Upang madaling matandaan kung ano - ano lamang ang POSITIVE SIGNS ng numeric values ng mga Trigonometric Functions, tandaan lamang ang phrase na ito: “All Stations To Central” at ang drawing sa ibaba




PAGSASANAY

A. Find the reference angle of each of the following angles. 

1. 135°
2. 150°
3. 210°
4. 300°
5. 315°

B.    Determine the values of the following functions. 

1. sin 135°
2. cos 210°
3. tan 240°
4. sec 315°
5. csc 330°

MGA SAGOT SA PAGSASANAY