Math Triangle Trig Day 1.2 Solution

Right Triangle Homework WS (Day 1) Express each angle in the opposite form. 72° 5π/5 -300° 5π/3 3960° Find the exact value of the six trigonometric ratios of the angle θ in the triangle. sinθ = 15/17 cosθ = 8/17 tanθ = 15/8 cscθ = 17/15 secθ = 17/8 cotθ = 8/15 sinθ = 7/8 cosθ = √15/8 tanθ = 7/√15 = 7√15/15 cscθ = 8/7 secθ = 8/√15 = 8√15/15 cotθ = √15/7 Sketch a triangle that has acute angle θ, and find the other five trigonometric ratios of θ. sinθ = 3/5 cosθ = 4/5 tanθ = 3/4 cscθ = 5/3 secθ = 5/4 cotθ = 4/3 cscθ = 13/12 sinθ = 12/13 cosθ = 5/13 tanθ = 12/5 secθ = 13/5 cotθ = 5/12 Find the EXACT length of the side labeled x. sin30° = x/25 x = 25/2 sin45° = 12/x x = 12√2 sin60° = x/13 x = 13√3/2 Evaluate the expression without using a calculator. sin(30°) + cos(30°) = 1/2 + √3/2 = (1+√3)/2 (sin(60°))² + (cos(60°))² = (√3/2)² + (1/2)² = 3/4 + 1/4 = 1 Solve the following right triangles. For #21, find the EXACT VALUES. For #24, you can use DECIMALS. MAKE SURE YOU FIND ALL MISSING SIDES AND ALL MISSING ANGLES. x = 16 y = 16√2 x = 35 tan(52°) ≈ 44.8 y = 35 / cos(52°) ≈ 56.8 x = 100√3 / 3 y = 200√3 / 3 x = 85√3 / 3 y = 170√3 / 3 x = 61.27 (approximate value) Right Triangle Homework WS (Day 2) Determine the measurements for angle θ for each of the following triangles. tanθ = 25/13 θ = tan⁻¹(25/13) ≈ 62.5° sinθ = 10/18 θ = sin⁻¹(10/18) ≈ 34.2° cosθ = 8/20 θ = cos⁻¹(8/20) ≈ 66.4° Applications: It might be helpful to draw a picture! Round all answers to the nearest tenth. A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5°. How tall is the tree?   tan(71.5°) = h/50 h = 50 * tan(71.5°) ≈ 149.4 feet A 30-meter line is used to tether a helium-filled balloon. Because of a breeze, the line makes an angle of approximately 75° with the ground. How high is the balloon off the ground?   sin(75°) = h/30 h = 30 * sin(75°) ≈ 28.98 m A safety regulation states that the maximum angle of elevation for a rescue ladder is 72°. A fire department's longest ladder is 110 feet. What is the maximum safe rescue height?   sin(72°) = h/110 h = 110 * sin(72°) ≈ 104.6 feet A ski slope on a mountain has an angle of elevation of 25.2°. The vertical height of the slope is 1808 feet. How long is the slope?   sin(25.2°) = 1808/s s = 1808 / sin(25.2°) ≈ 4246.3 ft 8. A lighthouse keeper observes that there is a 3° angle of depression to a ship in the ocean. If the keeper is 19 m above the water, how far is the ship from the base of the lighthouse? tan(87°) = x/19 x ≈ 362.5 m 9. At a point 200 feet from the base of a building, the angle of elevation to the bottom of a smokestack on the roof of the building is 35°, and the angle of elevation to the top of the smokestack is 53°. Find the height of the smokestack.   tan(35°) = x/200 x ≈ 140.04 tan(53°) = h/200 h ≈ 265.41 height of smokestack = h - x = 265.41 - 140.04 ≈ 125.37 ft 10. An airplane is flying at an elevation of 5150 feet above a straight highway. Two motorists are driving cars on the highway on opposite sides of the plane. The angle of depression to one car is 32° and the angle of depression to the other car is 52°. How far apart are the cars?   tan(32°) = 5150/x x ≈ 8241.72 tan(52°) = 5150/y y ≈ 4023.62 distance between cars = x + y ≈ 12265.34 ft 11. A student looks out of a second-story school window and sees the top of the school flagpole at an angle of elevation of 22°. The student is 18 ft above the ground and 50 ft from the flagpole. Find the height of the flagpole.   tan(22°) = x/50 x ≈ 20.2 height of flagpole = 20.2 + 18 = 38.2 ft 12. In traveling across flat land you notice a mountain directly in front of you. Its angle of elevation to the peak is 3.5°. After you drive 13 miles closer to the mountain, the angle of elevation is 9°. Approximate the height of the mountain.   tan(9°) = h/x tan(3.5°) = h/(x+13) h = x * tan(9°) = (x+13) * tan(3.5°) x * tan(9°) = x * tan(3.5°) + 13 * tan(3.5°) x(tan(9°) - tan(3.5°)) = 13 * tan(3.5°) x ≈ 8 h = 8 * tan(9°) ≈ 1.26 miles