Solving Right Triangles Lesson 14.3 Solving Right Triangles pp. 594-598

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

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

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° = 0.3090 = 0.1584 = 0.8572 = 1.8040

Find mA to the nearest degree given the following trig ratios. 5. tan A = 1.3250 6. cos A = 0.9455 7. sin A = 0.9130 mA = 53° mA = 19° mA = 66°

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.

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.

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.

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°

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

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°

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 ° =

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°

EXAMPLE 1 Given right ABC, find the measure of each side and angle. 8 24.6 c 72° 18° 82 + 24.62 = c2 64 + 605.16 = c2

EXAMPLE 1 Given right ABC, find the measure of each side and angle. 8 24.6 c 72° 18° 669.16 = c2 c  25.9

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

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

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

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

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

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

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.

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

EXAMPLE 3 Solve right ABC. 17 11 A = B = C = a = b = c = 90° 17 11 172 + 112 = c2 c2 = 410

EXAMPLE 3 Solve right ABC. 17 11 A = B = C = a = b = c = 90° 17 11 20.2 172 + 112 = c2 c  20.2

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°

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°

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

Homework pp. 597-598

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

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

►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 = 0.8746

►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°

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

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

►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

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

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

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

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

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

Analytic Geometry Measurement

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

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