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Trigonometric Functions in Terms of Opp, Adj, Hyp. DISCUSSION
Precalculus
So far we have expressed the trigonometric functions in terms of a circle
with radius r.
(x, y)
r
y
θ
x
In terms of a circle with radius r, our 6 trigonometric functions are:
sin  
______
______
cos 
______
______
tan  
______
______
csc 
______
______
sec 
______
______
cot 
______
______
Notice that in the diagram above we also have a right triangle with legs x,
y, and hypotenuse r. In terms of a right triangle we have another way of
expressing our trigonometric functions that will still give the same values,
but from a viewpoint of a right triangle rather than a circle. The right
triangle is enlarged below:
r
y
θ
x
With respect to the angle  , the side with y can be considered to be the
_________________ side, the side with x can be considered to be the
________________ side, and the radius is the ____________________.
Our six trigonometric functions can now be expressed as:
sin  
________
________
cos 
_______
_______
tan  
_________
_________
csc 
_________
_________
sec 
_________
_________
cot 
_________
_________
Notice that in a right triangle there are two angles that are not equal to
900. In the triangle below, these two angles are  and  .
B
α
c
a
θ
A
b
C
With respect to angle θ the opposite side has length ______, the adjacent
side has length ______, and the hypotenuse has length _____. The
trigonometric functions for angle θ, then, are:
sin  
____________ ___

____________ ___
csc 
____________ ___

____________ ___
cos 
___________ ___

___________ ___
sec 
____________ ___

____________ ___
tan  
_____________ ___

_____________ ___
cot 
____________ ___

____________ ___
With respect to angle α the opposite side has length ______, the adjacent
side has length ______, and the hypotenuse has length _____. The
trigonometric functions for angle θ, then, are:
sin  
____________ ___

____________ ___
csc 
____________ ___

____________ ___
cos 
___________ ___

___________ ___
sec 
____________ ___

____________ ___
tan  
_____________ ___

_____________ ___
cot 
____________ ___

____________ ___
Notice that the two acute angles in a right triangle are _______________
of one another (meaning they add up to ________). Also notice that for
complementary angles the sin of one angle is equal to the _________ of
its complement. We have seen this already in looking at the
trigonometric function values for two of our special angles, 300 and 600
(which are complements of one another). The sin 300 = _______ and the
cos 600 = _______. Also, the sin 600 = _______ and the cos 300 = ____.
Try this example:
For angle B: Opposite side = ______
B
Adjacent side = ________
5
3
Hypotenuse = _________
A
4
sin B 
_____
_____
cos B 
_____
_____
tan B 
_____
_____
For angle A: Opp side = _____ Adj side = ______ Hyp = ______
sin A 
_____
_____
cos A 
_____
_____
tan A 
_____
_____
If one of the acute angles of a right triangle is known and one side length
is known, trig functions can be used to determine the value for any other
side length. Try these examples:
opposite
adjacent
opposite
sin 
cos 
tan 
hypotenuse
hypotenuse
adjacent
Which trig function will be used? ____________
Solve for the unknown length.
30 
7
______ 30 0 
______
______
g
g  ______  ______  ______
c
75 50
Which trig function will be used? __________
Solve for the unknown length.
______
______
c  ______  ______
______
c 
 ______
______
______ 75 0 
Trigonometric functions are used in many real world problems where a
situation can be expressed in terms of a right triangle:
Example Word Problems
sin  
opposite
hypotenuse
cos  
adjacent
hypotenuse
tan  
opposite
adjacent
A plane rises at an angle of 10o with respect to the ground. Find the
height, h, above the ground the plane is after it has traveled 200 meters on
its path.
(1.) Draw a sketch of the situation. Label the angle, known length,
and unknown length symbol on the diagram.
(2.) Solve for the unknown height.
sin  
opposite
hypotenuse
cos  
adjacent
hypotenuse
tan  
opposite
adjacent
A board leans against a building and makes an angle of 40o with the
ground. If the end of the board is 4 meters from the building, what is the
length, l, of the board?
(1.) Draw a sketch of the situation. Label the angle, known length, and
unknown length symbol on the diagram.
(2.) Solve for the unknown length.
Inverse trigonometric functions can also be used to find an angle in a
word problem if two sides of a right triangle are known and you want to
determine an angle.
Example: You are standing 100 meters away from a building that is 170
meters tall. At what angle must you look up at to see the top
of the building?
(1.) Draw a sketch of the situation. Label the angle, known length,
and unknown length symbol on the diagram.
(2.) Solve for the unknown angle.
Trigonometric Functions in terms of Opp., Adj, and Hyp.
Precalculus
For each triangle, find the indicated trig function value. Express answers
as fractions in simplest form.
sin  
opposite
hypotenuse
cos 
adjacent
hypotenuse
tan  
opposite
adjacent
Hypotenuse
Opposite leg
θ
Adjacent leg
B
1.)
61
11
C
60
A
Sin A =
______
______
Sin B =
______
______
Cos A =
______
______
Cos B =
______
______
Tan A =
______
______
Tan B =
______
______
B
2.)
58
A
40
42
C
Tan A =
______
______
Sin A =
______
______
Cos B =
Sin B =
______
______
______
______
Tan B =
Cos A =
______
______
______
______
Hypotenuse
Opposite leg
θ
Adjacent leg
Fill in the blanks and determine the unknown side length using
trigonometry.
sin 
opposite
hypotenuse
cos 
adjacent
hypotenuse
tan 
opposite
adjacent
3.)
Which trig function will be used? ____________
Solve for the unknown length.
50
3
______ 50 0 
g
______
______
g  ______  ______  ______
4.)
d
28
5
Which trig function will be used? __________
Solve for the unknown length.
______
______
d  ______  ______  ______
______ 28 0 
5.)
4
75 s
Which trig function will be used? ____________
Solve for the unknown length.
6.)
58
h
8
Which trig function will be used? ____________
Solve for the unknown length.
Applications:
sin  
opposite
hypotenuse
cos  
adjacent
hypotenuse
tan  
opposite
adjacent
7.) Someone is flying a kite. The person lets out 40 meters of string.
The angle the string makes with the ground is 300. How high, h, is
the kite in the air?
(1.) Draw a sketch of the situation. Label the angle, known length,
and unknown length symbol on the diagram.
(2.) Solve for the unknown length.
8.) A ladder makes an angle of 600 with the ground. Its top extends 5 m
up the side of a building. What is the length, l, of the ladder?
(1.) Draw a sketch of the situation. Label the angle, known length,
and unknown length symbol on the diagram.
(2.) Solve for the unknown length.
sin  
opposite
hypotenuse
cos  
adjacent
hypotenuse
tan  
opposite
adjacent
9.) A street goes up at an angle of 50. What will the horizontal distance,
d, be between two points on the road that differ from one another by
10 meters in height?
(1.) Draw a sketch of the situation. Label the angle, known length,
and unknown length symbol on the diagram.
(2.) Solve for the unknown distance.
10.) The stairway in a home goes up at an angle of 360. What is the
length, l, of the stairway if the height gain in going up the stairs is 3
meters?
(1.) Draw a sketch of the situation. Label the angle, known length,
and unknown length symbol on the diagram.
(2.) Solve for the unknown stairway length, l.
opposite
adjacent
opposite
cos  
tan  
hypotenuse
hypotenuse
adjacent
11.) The wheelchair ramp going into a home is at an angle of 80 with
the ground. In total, the vertical height rise of the ramp is 1.3 m.
Horizontally, what is the distance, d, from the home where the
ramp starts?
(1.) Draw a sketch of the situation. Label the angle, known height,
and unknown distance symbol on the diagram.
sin  
(2.) Solve for the unknown distance, d.
12.) A surveyor wants to determine the height of a vertical cliff without
actually climbing it. She walks 85 meters away from the bottom of
the cliff and measures the angle between the ground and the top of
the cliff. If this angle is 52o, how high, h, is the cliff?
(1.) Draw a sketch of the situation. Label the angle, known height,
and unknown distance symbol on the diagram.
(2.) Solve for the unknown height, h.
13.) At a point on the ground 4000 meters from where a plane has lifted
off the plane is 300 meters in the air. At what angle in degrees is the
plane rising?
(1.) Draw a sketch of the situation.
(2.) Calculate the angle in degrees the plane is rising at.