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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. Your answers should be accurate to four decimal places. 1. cos 72° = 0.3090 2. tan 9° = 0.1584 3. sin 59° = 0.8572 4. tan 61° = 1.8040 Find mA to the nearest degree given the following trig ratios. 5. tan A = 1.3250 mA = 53° 6. cos A = 0.9455 mA = 19° 7. sin A = 0.9130 mA = 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. B c 72° 8 A A= B = 72° C = 90° b C a =8 b= c= EXAMPLE 1 Given right ABC, find the measure of each side and angle. B c 72° 8 A b A = 90° - B = 90° - 72° = 18° C EXAMPLE 1 Given right ABC, find the measure of each side and angle. B c 72° 8 A A = 18° B = 72° C = 90° b C a =8 b= c= EXAMPLE 1 Given right ABC, find the measure of each side and angle. B c 72° 8 18° A b C b tan 72 = 8 b= (tan 24 .6 72) 8 EXAMPLE 1 Given right ABC, find the measure of each side and angle. B c 72° 8 18° A A = 18° B = 72° C = 90° b C a =8 b = 24.6 c= EXAMPLE 1 Given right ABC, find the measure of each side and angle. B c 72° 8 18° A 24.6 82 + 24.62 = c2 64 + 605.16 = c2 C EXAMPLE 1 Given right ABC, find the measure of each side and angle. B c 72° 8 18° A 24.6 669.16 = c2 c 25.9 C EXAMPLE 1 Given right ABC, find the measure of each side and angle. B c 72° 8 A A = 18° B = 72° C = 90° b C a =8 b = 24.6 c = 25.9 EXAMPLE 2 Solve right ABC if C is the right angle, mA = 38°, and c = 26 units. B A = 38° B = 52° 26 C = 90° a= 38° b= A C c = 26 EXAMPLE 2 Solve right ABC if C is the right angle, mA = 38°, and c = 26 units. a B sin 38° = 26 a = sin 38°(26) a 16.0 26 38° A C EXAMPLE 2 Solve right ABC if C is the right angle, mA = 38°, and c = 26 units. B A = 38° B = 52° 26 C = 90° a = 16.0 38° b= A C c = 26 EXAMPLE 2 Solve right ABC if C is the right angle, mA = 38°, and c = 26 units. b B cos 38° = 26 b = cos 38°(26) b 20.5 26 38° A C EXAMPLE 2 Solve right ABC if C is the right angle, mA = 38°, and c = 26 units. B A = 38° B = 52° 26 C = 90° a = 16.0 38° b = 20.5 A C c = 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. A= A B= C = 90° 11 a = 17 b = 11 C 17 B c= 172 + 112 = c2 c2 = 410 EXAMPLE 3 Solve right ABC. A= A B= C = 90° 11 a = 17 b = 11 C 17 B c = 20.2 172 + 112 = c2 c 20.2 EXAMPLE 3 Solve right ABC. A = 57° A B= C = 90° 11 a = 17 b = 11 C 17 B c = 20.2 tan A = 17/11 tan-1 (17/11) = A 57° EXAMPLE 3 Solve right ABC. A = 57° A B = 33° C = 90° 11 a = 17 b = 11 C 17 B c = 20.2 B = 90° - 57° B = 33° EXAMPLE 3 Solve right ABC. A = 57° A B = 33° C = 90° 11 a = 17 b = 11 C 17 B c = 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 mA, 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. E 11. DF 20 17° F D ►B. Exercises Solve each right triangle. Round your answers to the nearest tenth or to the nearest X degree. 17. 15 Z 12 Y ►B. Exercises Solve each right triangle. Round your answers to the nearest tenth or to the nearest degree. M 19. L 26° 5 N ►B. Exercises Solve right ∆ABC if C is the right angle. Round your answers to the nearest tenth or to the nearest degree. 23. mA = 47°, b = 18 units ■ Cumulative Review Which are congruent, similar, or neither? Why? D 27. B E C A AB || CD ■ Cumulative Review Which are congruent, similar, or neither? Why? 28. P R Q S ■ Cumulative Review Which are congruent, similar, or neither? Why? 29. F G H E ■ Cumulative Review Which are congruent, similar, or neither? Why? M 30. I 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)