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
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