Practice and Problem-Solving Exercises

A Practice

Find each value without using a calculator. See Problem 1.

8. tan ( − π ) tangent open negative pi close
9. tan π tangent pi
10. tan 3 π 4 tangent . fraction 3 pi , over 4 end fraction
11. tan π 2 tangent , pi over 2
12. tan ( − 7 π 4 ) tangent . open . negative , fraction 7 pi , over 4 end fraction . close
13. tan 2 π tangent 2 pi
14. tan ( − 3 π 4 ) tangent . open . negative , fraction 3 pi , over 4 end fraction . close
15. tan ( 3 π 2 ) tangent . open , fraction 3 pi , over 2 end fraction , close

Each graphing calculator screen shows the interval 0 to 2π. What is the period of each graph? See Problem 2.

Identify the period and determine where two asymptotes occur for each function.

18. y = tan 5 θ y equals tangent 5 theta
19. y = tan 3 θ 2 y equals tangent . fraction 3 theta , over 2 end fraction
20. y = tan 4 θ y equals tangent 4 theta
21. y = tan 2 3 π θ y equals tangent . fraction 2 , over 3 pi end fraction , theta

Sketch the graph of each tangent curve in the interval from 0 to 2π.

22. y = tan θ y equals tangent theta
23. y = tan 2 θ y equals tangent 2 theta
24. y = tan 2 π 3 θ y equals tangent . fraction 2 pi , over 3 end fraction , theta
25. y = tan ( − θ ) y equals tangent open negative theta close

Graphing Calculator Graph each function on the interval 0 ≤ x ≤ 2 π  and  − 200 ≤ y ≤ 200 . 0 less than or equal to x less than or equal to 2 pi , and , minus 200 less than or equal to y less than or equal to 200 .  Evaluate each function at x = π 4 ,  π 2 , x equals , fraction pi , over 4 end fraction , comma , fraction pi , over 2 end fraction , comma  and 3 π 4 . fraction 3 pi , over 4 end fraction , .  See Problem 3.

26. y = 50 tan x y equals 50 tangent x
27. y = − 100 tan x y equals negative 100 tangent x
28. y = 125 tan ( 1 2 x ) y equals 125 tangent . open , 1 half , x , close
29. Graphing Calculator Suppose the architect in Problem 3 reduces the length of the base of the triangle to 100 ft. The function that models the height of the triangle becomes y = 50 tan θ . y equals 50 tangent theta .
    1. Graph the function on a graphing calculator.
    2. What is the height of the triangle when θ = 16 ° ? theta equals 16 degrees question mark
    3. What is the height of the triangle when θ = 22 ° ? theta equals 22 degrees question mark

B Apply

Identify the period for each tangent function. Then graph each function in the interval from − 2 π negative 2 pi  to 2 π . 2 pi .

30. y = tan π 6 θ y equals tangent , pi over 6 , theta
31. y = tan 2 . 5 θ y equals tangent 2 . 5 theta
32. y = tan ( − 3 2 π θ ) y equals tangent . open . negative , fraction 3 , over 2 pi end fraction , theta . close

Graphing Calculator Solve each equation in the interval from 0 to 2π. Round your answers to the nearest hundredth.

33. tan θ = 2 tangent theta equals 2
34. tan θ = − 2 tangent theta equals negative 2
35. 6 tan 2 θ = 1 6 tangent 2 theta equals 1
36. Open-Ended Write a tangent function.
37. Graph the function on the interval − 2 π negative 2 pi  to 2 π . 2 pi .
38. Identify the period and the asymptotes of the function.

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