1. Lesson 12.1
Right Triangle
Trigonometry

2. Your Notes

4. Keywords:
sine, cosine,
tangent, cosecant,
cotangent

5. Each of the six possible ratios of the three sides of a right triangle is used to define
a trigonometric function.

6. You will use an
airplane’s height and angle of
descent to find this distance.

8. q Evaluate Trigonometric Functions Example 1
Evaluate the six trigonometric functions of the angle shown in the triangle. q
SOLUTION
From the Pythagorean theorem, the length of the hypotenuse is,
5 2
122 + =
169 13
Using adj = 5, opp = 12, and hyp =13, you can write the following. 8

9. q q q q Evaluate Trigonometric Functions Example 1 = hyp opp sin 13 12
adj
cos q 13 5 =
adj
opp
tan q 5 12 =
opp
hyp
csc q 12 13
adj
sec 5
cot 9

10. Checkpoint
Evaluate Trigonometric Functions 1.
Evaluate the six trigonometric functions of the angle shown in the triangle. q
ANSWER , =
sin q 25 7
cos 24
tan
csc
cot
sec

13. Find a Missing Side Length
Example 2
Find a Missing Side Length
Find the value of x for the right triangle shown.
SOLUTION
Write an equation using a trigonometric function that involves the ratio of x and 8. Then solve the equation for x.
hyp
adj =
cos
30°
Write trigonometric equation. 8 x =
cos
30°
Substitute. 13

14. Find a Missing Side Length
Example 2
Find a Missing Side Length 3 x =
Use table to find 30º 2 8 x = 4 3
Multiply each side by 8.
ANSWER
The length of the longer leg is
6.93. 4 3 ≈ 14

15. Find an Unknown Side Length
Checkpoint
Find an Unknown Side Length
Find the value of x for the right triangle. 2.
ANSWER 4 3 3.
ANSWER 9 2 4. 13 2
ANSWER

16. q Use a Calculator Example 3
Find the value of x for the right triangle. a. b.
SOLUTION a. =
adj
opp
tan q b.
hyp
cos = 10 x
tan 25° 3
cos 63° 16

17. Example 3 Use a Calculator = x ( ) tan 25° 10 = 3 ( ) cos 63° x x 4.7≈ =
cos 63° 3 x
6.6 ≈ 17

18. q Use Trigonometry in Real-life Write trigonometric equation.
Example 4
Use Trigonometry in Real-life
Planes How far is the plane
from the landing spot?
SOLUTION
Write an equation using a trigonometric function that involves the ratio of x and 800. Then solve the equation for x. =
opp
hyp
csc q
Write trigonometric equation. =
800 x
csc 15°
Substitute. 18

19. Use Trigonometry in Real-life
Example 4
Use Trigonometry in Real-life =
800 x
sin 15° 1
Use the reciprocal of csc 15°.
Multiply each side by 800. x
3091 ≈
ANSWER
The plane is about 3091 feet from the landing spot. 19

20. Use Trigonometric Equations
Checkpoint
Use Trigonometric Equations 5.
Write a trigonometric equation to find the length of the hypotenuse in part (a) of Example 3.
ANSWER = y 10
cos 25°
sec 25° or
In Example 4, how far is the plane from the landing spot if the plane is flying at a height of 650 feet and is heading toward the landing spot at an angle of 20°? 6.
ANSWER
about 1900 ft

22. Suppose you are standing on the Grand View Terrace viewing platform at Mount Rushmore 550 feet from the base of the monument. You look up at the top of Mount Rushmore at an angle of 39°. How high is the top of the monument from where you are standing?

23. Slip