13-1 Practice. Trigonometric Identities. Find the exact value of each expression if 0 < θ < 90. 1, find sin θ. 1. If cos θ = 1, find cot θ.

Size: px

Start display at page:

Download "13-1 Practice. Trigonometric Identities. Find the exact value of each expression if 0 < θ < 90. 1, find sin θ. 1. If cos θ = 1, find cot θ."

Jasper Montgomery
5 years ago
Views:

1 1-1 Practice Trigonometric Identities Find the exact value of each expression if 0 < θ < If cos θ = 5 1, find sin θ.. If cot θ = 1, find sin θ.. If tan θ = 4, find sec θ. 4. If tan θ =, find cot θ. 5 Find the exact value of each expression if 180 < θ < If sin θ = , find sec θ. 6. If csc θ = -, find cot θ. Find the exact value of each expression if 70 < θ < If cos θ =, find cot θ. 8. If csc θ = -8, find sec θ If tan θ = - 1, find sin θ. 10. If cos θ = 1, find cot θ. Simplify each expression. 11. csc θ tan θ 1. sin θ tan θ 14. cot θ csc θ - cot θ 1 - cos θ 17. sin θ + cos θ cot θ 18. cos θ 1 - sin θ - cos θ 1 + sin θ 1. sin θ cot θ 0. AERIAL PHOTOGRAPHY The illustration shows a plane taking an aerial photograph of point A. Because the point is directly below the plane, there is no distortion in the image. For any point B not directly below the plane, however, the increase in distance creates distortion in the photograph. This is because as the distance from the camera to the point being photographed increases, the exposure of the film reduces by (sin θ)(csc θ - sin θ). Express (sin θ)(csc θ - sin θ) in terms of cos θ only. 16. csc θ - sin θ cos θ 19. sec θ cos θ + tan θ θ A B 1. WAVES The equation y = a sin θt represents the height of the waves passing a buoy at a time t in seconds. Express a in terms of csc θt. Chapter 1 8 Glencoe Algebra

2 1- Practice Verifying Trigonometric Identities 1. sin θ + cos θ cos θ = sec θ. cos θ 1 - sin θ = 1. (1 + sin θ)(1 - sin θ) = cos θ 4. tan 4 θ + tan θ + 1 = sec 4 θ 5. cos θ cot θ = cot θ - cos θ 6. (sin θ)(csc θ + sec θ) = sec θ 7. PROJECTILES The square of the initial velocity of an object launched from the ground is v = gh, where θ is the angle between the ground and the initial path h is the sin θ maximum height reached, and g is the acceleration due to gravity. Verify the identity gh sin θ = gh sec θ sec θ LIGHT The intensity of a light source measured in candles is given by I = ER sec θ, where E is the illuminance in foot candles on a surface, R is the distance in feet from the light source, and θ is the angle between the light beam and a line perpendicular to the surface. Verify the identity ER (1 + tan θ) cos θ = ER sec θ. Chapter 1 14 Glencoe Algebra

3 1- Practice Sum and Difference of Angles Identities Find the exact value of each expression. 1. cos 75. cos 75. sin (-165 ) 4. sin (-105 ) 5. sin cos sin 5 8. sin (-75 ) 9. sin cos (180 - θ) = -cos θ 11. sin (60 + θ) = sin θ 1. sin (45 + θ) - sin (45 - θ) = sin θ 1. cos ( x - π 6 ) + sin ( x - π ) = sin x 14. SOLAR ENERGY On March 1, the maximum amount of solar energy that falls on a square foot of ground at a certain location is given by E sin (90 - ϕ), where ϕ is the latitude of the location and E is a constant. Use the difference of angles formula to find the amount of solar energy, in terms of cos ϕ, for a location that has a latitude of ϕ. 15. ELECTRICITY In a certain circuit carrying alternating current, the formula c = sin (10t) can be used to find the current c in amperes after t seconds. a. Rewrite the formula using the sum of two angles. b. Use the sum of angles formula to find the exact current at t = 1 second. Chapter 1 0 Glencoe Algebra

4 1-4 Practice Double-Angle and Half-Angle Identities Find the exact values of sin θ, cos θ, sin θ, and cos θ for each of the following. 1. cos θ = 5 1, 0º < θ < 90º. sin θ = 8, 90º < θ < 180º 17. cos θ = 1 4, 70º < θ < 60º 4. sin θ = -, 180º < θ < 70º Find the exact value of each expression. 5. tan 105º 6. tan 15º 7. cos 67.5º 8. sin ( - π 8 ) 9. sin θ = tan θ - sin θ tan θ 10. sin 4θ = 4 cos θ sin θ cos θ 11. AERIAL PHOTOGRAPHY In aerial photography, there is a reduction in film exposure for any point X not directly below the camera. The reduction E θ is given by E θ = E 0 cos 4 θ, where θ is the angle between the perpendicular line from the camera to the ground and the line from the camera to point X, and E 0 is the exposure for the point directly below the camera. Using the identity sin θ = 1 - cos θ, verify that E 0 cos 4 θ = E 0 ( 1 + cos θ ). 1. IMAGING A scanner takes thermal images from altitudes of 00 to 1,000 meters. The width W of the swath covered by the image is given by W = H tan θ, where H is the H sin θ height and θ is half the scanner s field of view. Verify that = H tan θ. 1 + cos θ Chapter 1 6 Glencoe Algebra

5 1-5 Practice Solving Trigonometric Equations Solve each equation for the given interval. 1. sin θ = cos θ, 90º θ < 180º. cos θ = sin θ, 0º θ, 60º. cos 4θ = cos θ, 180º θ < 60º 4. cos θ + cos (90 - θ) = 0, 0 θ < π 5. + cos θ = sin θ, π θ π 6. tan θ + sec θ = 1, π θ < π Solve each equation for all values of θ if θ is measured in radians. 7. cos θ = sin θ 8. cot θ = cot θ 9. sin θ = sin θ 10. cos θ sin θ = sin θ 11. cos θ = 1 - sin θ 1. sec θ = Solve each equation for all values of θ if θ is measured in degrees. 1. sin θ cos θ = cos θ 14. csc θ - csc θ + = cos θ = 4(1 - cos θ) 16. cos θ = cos θ Solve each equation sin θ = sin θ - 1 = sin θ - sin θ = cos θ + sin θ - 1 = 0 1. WAVES Waves are causing a buoy to float in a regular pattern in the water. The vertical position of the buoy can be described by the equation h = sin x. Write an expression that describes the position of the buoy when its height is at its midline.. ELECTRICITY The electric current in a certain circuit with an alternating current can be described by the formula i = sin 40t, where i is the current in amperes and t is the time in seconds. Write an expression that describes the times at which there is no current. Chapter 1 Glencoe Algebra

Multiple-Angle and Product-to-Sum Formulas

Multiple-Angle and Product-to-Sum Formulas MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 011 Objectives In this lesson we will learn to: use multiple-angle formulas to rewrite