Download Right Triangle Trigonometry (All Six Functions)

Example 1 Evaluate Trigonometric Functions
Find the values of the six trigonometric functions for angle X.
leg opposite X: YZ = 3, leg adjacent X: XY = 2, and hypotenuse: XZ = 13
sin X =
csc X =
opp
hyp
hyp
opp
=
=
3
13
13
3 13
or
13
cos X =
sec X =
3
adj
hyp
hyp
adj
=
=
2
13
13
or
2 13
13
tan X =
cot X =
2
opp
adj
adj
opp
3
=
2
2
=
3
Example 2 Find Trigonometric Ratios
7
. Find the value of tan B.
12
Draw a right triangle and label one acute angle B. Since cos
7
adj
B
B=
=
, label the adjacent side 7 and the hypotenuse
12 hyp
12.
In a right triangle, B is acute and cos B =
Step 1
7
12
Step 2
b
Use the Pythagorean Theorem to find a.
a2 + b2 = c2
72 + b2 = 122
49 + b2 = 144
b2 = 95
b = ± 95
b=
Step 3
C
95
Find tan B.
opp
tan B = adj
=
95
7
A
Pythagorean Theorem
a = 7 and c = 12
Simplify.
Subtract 49 from each side.
Take the square root of each side.
Length cannot be negative.
Tangent function
Replace opp with
95 and adj with 7.
Example 3 Find a Missing Side Length
Use a trigonometric function to find the value of x. Round to the nearest
tenth if necessary.
The measure of the side opposite the angle is 25, the given angle measures
45˚, and the missing length is the hypotenuse of the triangle. The
trigonometric function relating the opposite side of a right triangle and the
hypotenuse is the sine function.
sin  =
sin 45˚ =
2
2
=
opp
x
45
25
sine ratio
hyp
25
Replace  with 45˚, opp with 25, and hyp with x.
x
25
2
2
Find the cross products.
sin 45˚ =
x
x 2 = 50
x=

50
Divide each side by
2
x = 25 2
2.
Simplify.
The value of x is 25 2 or about 35.4.
Example 4 Find a Missing Side Length
The Great Pyramid of Khufu in Egypt is one of the Seven Wonders of the World. Its
original height was 481 feet. The angle that the sides make with the square base is about
52˚. What is the length of one side of the base?
Make a drawing of the situation. The base will be twice the
length of a shown in the diagram. Write an equation using a
trigonometric function that involves the ratio of the length a
and 481.
tan 52˚ =
a=
481
a
481
tan  =
481 ft
opp
adj
Solve for a.
tan 52
a ≈ 375.8
Use a calculator.
To find the length of a side of the base, find 2a.
2a = 2(375.8) = 751.6
The length of one side of the base of the pyramid is about 752 feet.
52
a
Example 5 Find A Missing Angle Measure
Solve ∆RST. Round measures of sides to the nearest tenth and measures of angles to the
nearest degree.
20
T
You know the measures of the sides. You need to find S and R.
Find S.
20
2 5
opp
sin S =
or
sin S =
5
10 5
hyp
10 5
S
Use a calculator and the sin–1 function to find the angle whose sine is
KEYSTROKES: ` [SIN–1] 2 * ` [
] 5 )/5 )
To the nearest degree, S ≈ 63˚.
Find R.
63˚ + R = 90˚
R = 27˚
Angles R and S are complementary.
Solve for R.
Therefore, S ≈ 63˚ and R ≈ 27˚.
R
2 5
5
e 63.43494882
.
Example 6 Use Angles of Elevation and Depression
a. The world’s highest waterfall is Angel Falls in Venezuela. It is 3212 feet tall. If a tourist
estimates the angle of elevation to the top of the falls to be 35˚, what is the tourist’s
distance from the base of the falls?
Make a drawing of the situation. Let d be the tourist’s
distance from the base of the falls. Write an equation using a
trigonometric function that involves the ratio of d and 3212.
tan 35˚ =
d=
3212
tan  =
d
3212
opp
adj
Solve for d.
tan 35
d ≈ 4587.2
Use a calculator.
The tourist’s distance from the base of the waterfall is about 4587 feet.
b. A golfer is standing at the tee, looking up to the green on a
hill. If the tee is 28 yards lower than the green and the angle
of elevation from the tee to the hole is 18, find the distance
from the tee to the hole.
x yd
18
28 yd
Write an equation using a trigonometric function that involves the ratio of the vertical rise
(side opposite the 18 angle) and the distance from the tee to the hole (hypotenuse).
sin 18 =
28
x
x sin 18= 28
x=
28
sin18
opp
sin θ = hyp
Multiply each side by x.
Divide each side by sin 18.
x  58.25
Use a calculator.
So, the distance from the tee to the hole is about 58.3 yards.