Given one trigonometric ratio and quadrant, determining the remaining function values

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1 MATH 2412 Precalculus Sections Trigonometry (quick review) Below is a list of topics you should be familiar with if you have completed a course recently in Trigonometry. I am going to assume knowledge of these topics, so please ask questions next week over anyhting you might be concerned about. 4.1 Degree measure vs Radian measure Bearing Arc Length Angular Speed Linear Speed (along the circumference of a circle) 4.2 Right Triangle Definitions Special Angles Solving Right Triangles 4.3 Circular Function Definitions Standard Position Evaluating Trigonometric Functions Given one trigonometric ratio and quadrant, determining the remaining function values Trigonometric functions of real numbers The Unit Circle Periodic Functions Graphs of Sine and Cosine Sinusoidals

2 Translations/Transformations 4.5 Graphs of the remaining trigonometric functions Selected Examples: 1. Give the definitions for the six trigonometric functions, with respect to the angle B in the given right triangle C a B b c A 2. Using the figure shown, where the angle θ is in standard position in a rectangular coordinate system and the point P(x,y) is on the terminal side of θ, define the six trigonometric functions of θ. 3. Without using a calculator, determine the exact values of the six trigonometric functions of the following special angles. a) 60 b) 135 c) 180 d) 210

3 4. Give the reference angle, then a positive angle and a negative angle coterminal with each of the following: a) 1875 b) Given that tan β = 12 5 and sec β > 0, determine the values of the remaining trigonometric functions of β 6. An angle θ is in standard position on a rectangular coordinate system. Find the quadrant that θ lies in if: sec θ < 0 and sin θ > 0 7. Determine the values of the six trigonometric functions of θ, assuming that the terminal side of θ lies in quadrant III on the line x - 3y = An airplane takes off at an angle of 10 degrees and travels at the rate of 250 ft/sec. Approximately how long does the airplane take to reach an altitude of 28,000 feet?

4 9. A prop plane leaves the airport traveling at 215 mph at a heading of at the same time a jet plane leaves the airport traveling at 480 mph at a heading of Find the distance between them after two hours and thhe bearing of the prop plane from the jet plane. 10. From the sun deck of the Whalewatcher's Resort at Kehei, Maui, an observer watches a whale moving directly toward the resort. If the observer is 200 feet above the water and if the angle of depression from the observer to the whale changes from 15 to 32 during the period of observation, approximate the distance that the whale travels. 11. Evaluate the following expression without using a calculator. 2 2 sin cos270 sec tan Convert each degree measure to radians: (a) 225 (b) Find the exact value of the following: (a) ( ) tan π (b) 4π sin 3

5 14. Find the exact value(s) of s in the interval [ 0, 2π ] if 1 cos s =. 2 π 15. Sketch one period of the graph y = tan x 4 on the system below or the graph paper provided. Find the exact value of the y-intercept, if it exists. 16. Sketch one period of the graph of y 2cos( x π ) = +, clearly labeling both axes.use the graph paper provided or the system below. What is the exact value of the y-intercept, if it exists? What are the maximum and minimum values for y? π 17. Sketch one period of the graph y = 3csc x 2 on the system below or the graph paper provided. What is the exact value of the y-intercept, if it exists? 18. Without actually graphing the function defined by f( x) = 4 3sin 2( x+ π ) 6, write a detailed explanation of the transformations/translations needed to graph f( x ) from the graph of y = sin x.

6 19. Find two possible equations for the graph below. 20. Write an equation for a cotangent function with a period of π and a phase shift of 4 π to the left. 21. Write the domain and range for each circular function. Sin s Cos s Tan s Sec s Csc s Cot s Domain Range 22. An electric hoist in the shape of a circle is being used to lift an underground cable. The radius of the hoist is 2.5 feet and the cable must be raised 26 feet in order to make a repair. Use arc length to determine the number of degrees through which the hoist must rotate.

7 23. Dallas, Texas is on a latitude of N, while Omaha, Nebraska is on a latitude of N. Find the distance between the cities. (use 6400 kilometers as the radius of Earth) 24. A satellite circles the earth at a height of 100 miles. If it circles the earth every 90 minutes, what is the angular velocity of the satellite? What is the linear velocity in miles per hour? (use 4000 miles as the radius of the earth) 25. The two pulleys in the figure have radii of 15 cm and 8 cm, respectively. The larger pulley rotates 25 times in 6 seconds. Find the number of rotations per second of the smaller pulley and the linear speed of the belt connecting them The full moon subtends an angle of 05. when viewed from the earth's surface. If the moon is approximately 240,000 miles away from earth, appropximate its diameter using the formula for arc length. 27. The pedals on a bicycle turn the front sprocket at 20 radians per second. The sprocket has a diameter of 10 inches. The back sprocket, connected to the wheel, has a diameter of 6 inches. Find the linear velocity of the chain and the angular velocity of the back sprocket. How fast is the bicycle traveling in mph if the radius of the back wheel is 15 inches?