Math Review 6 Solution Key

Pre-Calculus Convert from radians to degrees or from degrees to radians. 1. 72° * (π/180°) = 2π/5 2. -120° * (π/180°) = -2π/3 3. 5π/6 * (180°/π) = 150° 4. -11π/3 * (180°/π) = -600° Find two positive angles and two negative angles that are coterminal with the given angle. 5. 50° 50° + 360° = 410° 50° + 360° = 770° 50° - 360° = -310° 50° - 360° = -670° 6. -π/4 -π/4 + 2π = 7π/4 -π/4 + 2π = 15π/4 -π/4 - 2π = -9π/4 -π/4 - 2π = -17π/4 7. Find an angle between 0° and 360° that is coterminal with -800°. -800° + 360°(3) = 280° 8. Find an angle between 0 and 2π that is coterminal with -17π/4. -17π/4 + 24π/4 = 7π/4 9. Find the length of arc s in the figure. s = rθ = 4(120° * π/180°) = 8π/3 10. An arc of length 60m subtends a central angle in a circle of radius 20m. Find θ in radians. 60 = 20θ θ = 3 11. A sector of a circle has a central angle of 62°. Find the area of the sector if the radius of the circle is 3 mi. (3)² * (62°/180°) = 3π/15 12. The area of a sector of a circle is 16 m². If the central angle is 2 radians, find the radius of the circle. 16 = 1/2 * r² * 2 r = 4 m 13. The wheels of a car have a radius of 11 inches and rotate at 600 rpm. Find the speed of the car in mi/hr. (600 rev/min) * (60 min/hr) * (2π * 11 in/rev) * (1 ft/12 in) * (1 mile/5280 ft) = 39.3 mi/hr 14. A truck has 48-inch diameter wheels that rotate at 520 rpm. a) Find the angular speed of the wheels in rad/min. 520 rev/min * 2π rad/rev = 1040π rad/min b) Find the linear speed of the wheels in mi/hr. 1040π rad/min * (60 min/hr) * (24 in/2π rad) * (1 ft/12 in) * (1 mile/5280 ft) = 74.3 mi/hr 15. Find the exact values of the six trig functions for angle θ in the triangle below. θ |\ | \ | \ 3 | \ 5 | \ |_ _ _ _ _ _ _ _ _ _ _ _ 2 sin θ = 2/5 csc θ = 5/2 cos θ = 3/5 sec θ = 5/3 tan θ = 2/3 cot θ = 3/2 Find the value of x. Use exact values when possible. 16. 30° |\ | \ | \ x | \ 32 | \ |_ _ _ _ _ _ _ _ _ _ _ _ sin 30° = x/32 x = 16 17. θ |\ | \ | \ x | \ 20 | \ |_ _ _ _ _ _ _ _ _ _ _ _ 62° sin 62° = 20/y y = 22.65 cos 62° = x/22.65 x = 24.43 18. Sketch a triangle that has an acute angle θ. If cot θ = 1, find the values of the other five trig functions for angle θ. θ |\ | \ | \ 1 | \ 1 | \ |_ _ _ _ _ _ _ _ _ _ _ _ sin θ = 1/√2 = √2/2 cos θ = 1/√2 = √2/2 csc θ = √2 sec θ = √2 tan θ = 1 19. Evaluate the expression without using a calculator: sin 30° * csc 30° (1/2) * (2/1) = 1 Solve the right triangle. 20. B |\ | \ | \ C | \ 30 | \ |_ _ _ _ _ _ _ _ _ _ _ _ 45° A x cos 45° = 30/y y = 30/cos 45° = 42.43 tan 45° = x/30 x = 30 * tan 45° = 30 A = 90° B = 45° C = 45° a = 42.43 b = 33.32 c = 30 21. B |\ | \ | \ C | \ 106 | \ |_ _ _ _ _ _ _ _ _ _ _ _ π/3 A x sin π/3 = 106/y y = 106/sin π/3 = 180.34 tan π/3 = 106/x x = 145.90 A = π/2 B = π/3 C = π/6 a = 180.34 b = 145.90 c = 106 22. A child is on a beach flying a kite. He holds the end of the kite string at ground level and estimates the angle of elevation of the kite to be 50°. If the string is 450 feet long,   then how high is the kite above ground? sin 50° = x/450 x = 450 sin 50° = 344.72 ft 23. When the moon is exactly half full, the earth, moon, and sun form a right angle. At that time, the angle formed by the sun, earth, and moon is measured to be 89.85°.   If the distance from the earth to the moon is 240,000 miles, estimate the distance from the earth to the sun.   cos 89.85° = 240000/x x = 240000/cos 89.85° ≈ 91,673,351.94 miles 24. From the top of a 150-ft lighthouse, the angle of depression to a ship in the ocean is 35°. How far is the ship from the base of the lighthouse?   tan 55° = x/150 x = 150 tan 55° ≈ 214.22 ft Find the reference angle for the given angle. (Write your answers in the form given!!) 25. 225° 45° 26. 810° 90° 27. -105° 75° 28. 3π/4 π/4 29. 4π/3 π/3 30. -5π/6 π/6 Find the EXACT VALUE of the trig function. SHOW YOUR WORK!!! 31. sin 225° = -√2/2 32. cos(-60°) = 1/2 33. tan(750°) = √3/3 34. tan(5π/6) = -√3/3 35. csc(-π/4) = -√2 36. sec(-2π/3) = -2 Find the quadrant in which θ lies given the following information. 37. tan θ < 0 and sin θ < 0 III 38. csc θ > 0 and cos θ < 0 II Find the values of the remaining five trig functions of θ given the following information. 39. csc θ = 2, θ is in Quadrant I sin θ = 1/2 cos θ = √3/2 tan θ = 1/√3 = √3/3 sec θ = 2/√3 = 2√3/3 cot θ = √3 40. cot θ = 1/2, sin θ < 0 sin θ = -2/√5 = -2√5/5 cos θ = -1/√5 = -√5/5 tan θ = 2 csc θ = -√5/2 sec θ = -√5 41. A triangle has an area of 16 in², and two sides of the triangle have lengths 5 in. and 7 in. Find the angle θ included by these two sides. 16 = 1/2 * 5 * 7 * sin θ sin θ = 16 / (5 * 7 / 2) = 66.1° 42. Find the area of the shaded region in the figure. Area of sector - Area of triangle = 1/2 * (4)² * (150/180) - 1/2 * (4) * (4) * sin 150° = 16.94 units²