1. Solving Right Triangles
Lesson 14.3
Solving Right Triangles pp

2. Objectives:
1. To find the trigonometric ratios for angles using a calculator or tables.
2. To find missing sides or angles of right triangles.

3. Solving a right triangle means finding all the angle measures and all the side lengths of the triangle from the information given.

4. Find the following using a calculator
Find the following using a calculator. Your answers should be accurate to four decimal places.
1. cos 72°
2. tan 9°
3. sin 59°
4. tan 61° = = = =

5. Find mA to the nearest degree given the following trig ratios.
5. tan A =
6. cos A =
7. sin A =
mA = 53°
mA = 19°
mA = 66°

6. There are two types of right triangles to solve.
1. The right triangle given a side and an acute angle.
2. The right triangle given two sides.

7. Steps to solve a right triangle given a side and an acute angle.
1. Find the other acute angle by subtracting the one given from 90°.
2. Set up a trig equation involving the acute angle and side given, and one of the unknown remaining sides.

8. Steps to solve a right triangle given a side and an acute angle.
3. Use the Pythagorean theorem and the two known sides to find the third.

9. EXAMPLE 1 Given right ABC, find the measure of each side and angle.8 b c
72°
A =
B =
C =
a =
b =
c = 8
72°
90°

10. EXAMPLE 1 Given right ABC, find the measure of each side and angle.8 b c
72°
A = 90° - B
= 90° - 72°
= 18°

11. EXAMPLE 1 Given right ABC, find the measure of each side and angle.8 b c
72°
A =
B =
C =
18°
a =
b =
c = 8
72°
90°

12. EXAMPLE 1 Given right ABC, find the measure of each side and angle.8 b c
72°
18° 8 b 72
tan = 6 . 24 b  ) 72
(tan 8 b =

13. EXAMPLE 1 Given right ABC, find the measure of each side and angle.8 b c
72°
18°
A =
B =
C =
18°
a =
b =
c = 8
72°
24.6
90°

14. EXAMPLE 1 Given right ABC, find the measure of each side and angle.8
24.6 c
72°
18°
= c2
= c2

15. EXAMPLE 1 Given right ABC, find the measure of each side and angle.8
24.6 c
72°
18°
= c2
c  25.9

16. EXAMPLE 1 Given right ABC, find the measure of each side and angle.8 b c
72°
A =
B =
C =
18°
a =
b =
c = 8
72°
24.6
90°
25.9

17. EXAMPLE 2 Solve right ABC if C is the right angle, mA = 38°, and c = 26 units.
52°
90° 26

18. EXAMPLE 2 Solve right ABC if C is the right angle, mA = 38°, and c = 26 units.
sin 38° = a 26 A B C 26
38°
a = sin 38°(26)
a  16.0

19. EXAMPLE 2 Solve right ABC if C is the right angle, mA = 38°, and c = 26 units.
52°
90°
16.0 26

20. EXAMPLE 2 Solve right ABC if C is the right angle, mA = 38°, and c = 26 units.
cos 38° = b 26 A B C 26
38°
b = cos 38°(26)
b  20.5

21. EXAMPLE 2 Solve right ABC if C is the right angle, mA = 38°, and c = 26 units.
52°
90°
16.0
20.5 26

22. Steps to solve a right triangle given two sides.
1. Find the third side using the given sides and the Pythagorean theorem.
2. Use the two given sides to set up a trig equation to find one of the acute angles.

23. Steps to solve a right triangle given two sides.
3. Subtract the acute angle from 90° to find the final angle.

24. EXAMPLE 3 Solve right ABC.17 11
A =
B =
C =
a =
b =
c =
90° 17 11
= c2
c2 = 410

25. EXAMPLE 3 Solve right ABC.17 11
A =
B =
C =
a =
b =
c =
90° 17 11
20.2
= c2
c  20.2

26. EXAMPLE 3 Solve right ABC.17 11
A =
B =
C =
a =
b =
c =
57°
90° 17 11
20.2
tan A = 17/11
tan-1 (17/11) = A  57°

27. EXAMPLE 3 Solve right ABC.17 11
A =
B =
C =
a =
b =
c =
57°
33°
90° 17 11
20.2
B = 90° - 57°
B = 33°

28. EXAMPLE 3 Solve right ABC.17 11
A =
B =
C =
a =
b =
c =
57°
33°
90° 17 11
20.2

29. Homework pp

30. ►A. Exercises
Find the indicated trigonometric ratios. See the table on p. 616.
1. sin 41°

31. ►A. Exercises
Find the indicated trigonometric ratios. See the table on p. 616.
3. tan 82°

32. ►A. Exercises
Find mA, given the following trigonometric ratios. See the table on page 616. Find the angles to the nearest degree.
7. cos A =

33. ►B. Exercises
Use the triangles shown. Name the ratio or theorem that you would use to find the indicated measurement and then calculate it.
11. DF E F D 20
17°

34. ►B. Exercises
Solve each right triangle. Round your answers to the nearest tenth or to the nearest degree.
17. X Z Y 15 12

35. ►B. Exercises
Solve each right triangle. Round your answers to the nearest tenth or to the nearest degree.
19. M L N
26° 5

36. ►B. Exercises
Solve right ∆ABC if C is the right angle. Round your answers to the nearest tenth or to the nearest degree.
23. mA = 47°, b = 18 units

37. ■ Cumulative Review Which are congruent, similar, or neither? Why? 27.D B E C A
AB || CD

38. ■ Cumulative Review Which are congruent, similar, or neither? Why? 28.P R S Q

39. ■ Cumulative Review Which are congruent, similar, or neither? Why? 29.F G E H

40. ■ Cumulative Review Which are congruent, similar, or neither? Why? 30.J K L N

41. ■ Cumulative Review Which are congruent, similar, or neither? Why? 31.Z W X Y V

42. Analytic Geometry
Measurement

43. ►Exercises Use the figure for exercises 1-2.
1. Find the perimeter of the triangle.
A (3, 5)
B (-2, 1)
C (1, -1)

44. ►Exercises Use the figure for exercises 1-2.
2. Find the area of the triangle.
A (3, 5)
B (-2, 1)
C (1, -1)