Right Triangle Trig Test Review Solution Key

Section 1: Converting Radians to Degrees and Vice Versa 4π/9 radians = 80° 2π/3 radians = 120° 14π/18 radians = 140° 11π/2 radians = 990° Section 2: Finding Trigonometric Functions Given a right triangle with sides 3, 4, and 5: sin(θ) = 3/5 cos(θ) = 4/5 tan(θ) = 3/4 csc(θ) = 5/3 sec(θ) = 5/4 cot(θ) = 4/3 Given sin(θ) = 5/6: cos(θ) = √11/6 tan(θ) = 5/√11 csc(θ) = 6/5 sec(θ) = 6/√11 cot(θ) = √11/5 Section 3: Finding Exact Values In a right triangle with sides 5 and 10: x = √(10² - 5²) = 5√3 In a right triangle with sides 9 and 45°: x = 9/sin(45°) = 9√2/2 In a right triangle with sides 14 and 60°: x = 14/sin(60°) = 28√3/3 Section 4: Evaluating Trigonometric Expressions (sin(45°))² + (cos(45°))² = 1 (tan(60°))² + (csc(45°))² = 5 sin(π/2) + cos(π/2) = 1 Section 5: Solving Triangles Given a right triangle with sides 4√6 and 45°: x = 4√6 y = 4√6 * √2 = 4√12 = 8√3 Given a right triangle with sides 38 and 30°: x = 38/sin(30°) = 76 y = 38/tan(30°) = 38√3/3 Section 1: Finding Side Lengths 15. Given a right triangle with angles 30°, 60°, and 90°, and one side length of 24: x = 24√3 y = 24 16. Given a right triangle with angles 45°, 45°, and 90°, and one side length of 32: x = 32√2 / 2 = 16√2 y = 32√2 / 2 = 16√2 17. Given a right triangle with angles 30°, 60°, and 90°, and one side length of 80: x = 80 / √3 = 80√3 / 3 y = 80 / 2 = 40 Section 2: Finding Angles 18. Given a right triangle with sides 6, 13, and 12: tan(θ) = 12/5 θ = tan⁻¹(12/5) ≈ 67.4° 19. Given a right triangle with sides 6, 8, and 10: cos(θ) = 6/10 = 3/5 θ = cos⁻¹(3/5) ≈ 53.1° 20. Given a right triangle with sides 8, 15, and 17: tan(θ) = 15/8 θ = tan⁻¹(15/8) ≈ 61.9° Section 3: Application 21. An airplane is flying at an elevation of 6500 feet above a highway. The pilot spots two cars on the highway, one at a 23° angle of depression and the other at an 18° angle of depression. triangle with sides and angles labeled tan(18°) = 6500 / w w = 6500 / tan(18°) ≈ 20004.9 tan(23°) = 6500 / y y = 6500 / tan(23°) ≈ 15313.0 x = w - y ≈ 4691.9 feet Problem 22: Given: A treehouse 20 feet above the ground and a park bench 40 feet from the base of the tree. Task: Find the angle of depression from the treehouse to the bench. Solution: tan(θ) = 20/40 θ = 26.6° (angle of depression) Problem 23: Given: A woman on a hill, a flagpole 60 feet tall, angle of depression to the bottom of the pole is 16°, and angle of elevation to the top is 20°. Task: Find the woman's distance from the flagpole. Solution: tan(20°) = x / y tan(16°) = (60 - x) / y Solving the system of equations: x ≈ 33.55, y ≈ 92.2 feet Problem 24: Given: A student 21 feet above the ground, 75 feet from a flagpole, and an angle of elevation to the top of the flagpole of 27°. Task: Find the height of the flagpole. Solution: tan(27°) = x / 75 x ≈ 38.21 feet Total height of the flagpole = 38.21 + 21 = 59.21 feet Problem 25: Given: A skateboard ramp 4 feet high with a 18.4° angle of elevation. Task: Find the length of the ramp. Solution: sin(18.4°) = 4 / L L = 4 / sin(18.4°) ≈ 12.7 feet Problem 26: Given: An escalator 152 feet long with a 30° angle of elevation. Task: Find the height of the platform. Solution: sin(30°) = x / 152 x = 152 * sin(30°) = 76 feet Problem 27: Given: Two points A and B 12 meters apart, angles of elevation to a tower from A and B are 35° and 45°, respectively. Task: Find the height of the tower. Solution: tan(45°) = h / x tan(35°) = h / (x + 12) Solving the system of equations: h ≈ 28 feet Problem 28: Given: A mountain with two angles of elevation: 6.5° and 14°, and a distance of 16 miles between the two observation points. Task: Find the height of the mountain. Solution: Using similar triangles and trigonometric ratios, the solution involves setting up equations and solving for the height (h). The final answer is approximately 3.4 miles. Problems 29-32: These problems involve determining the sign of trigonometric expressions in different quadrants. The answers are based on the definitions of trigonometric functions and the signs of sine, cosine, and tangent in each quadrant. Problems 33-36: Given the signs of trigonometric functions, these problems ask to determine the quadrant in which the angle θ lies. The solutions are based on the CAST rule, which helps remember the signs of sine, cosine, and tangent in each quadrant. Problems 37-40: Given a trigonometric function value and the quadrant, these problems ask to find the values of the other five trigonometric functions. The solutions involve using Pythagorean identities and the definitions of trigonometric functions.