13-5 The Cosine Function

Quick Review

The cosine function y = cos θ y equals cosine theta  matches the measure θ of an angle in standard position with the x-coordinate of a point on the unit circle. This point is where the terminal side of the angle intersects the unit circle.

For the cosine function y = a cos b θ , y equals eh cosine b theta comma  the amplitude equals | a |, there are b cycles from 0 to 2π, and the period is 2 π b . fraction 2 pi , over b end fraction , .

Example

Find all solutions to 5 cos θ = − 4 5 cosine theta equals negative 4  in the interval from 0 to 2π. Round each answer to the nearest hundredth.

On a graphing calculator graph the equations y = − 4 y equals negative 4  and y = 5 cos θ . y equals 5 cosine theta .

Use the Intersect feature to find the points at which the two graphs intersect. The graph shows two solutions in the interval. They are θ ≈ 2 . 50  and  3 . 79 theta almost equal to 2 . 50 , and , 3 . 79

Exercises

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

23. y = 2 cos ( π 2 θ ) y equals 2 cosine . open , pi over 2 , theta , close
24. y = − cos 2 θ y equals negative cosine 2 theta
25. Write an equation of a cosine function with a > 0 , eh greater than 0 comma  amplitude 3, and period π.

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

26. 3 cos 4 θ = − 2 3 cosine 4 theta equals negative 2
27. cos ( π θ ) = − 0.6 cosine open pi theta close equals negative 0.6

13-6 The Tangent Function

Quick Review

The tangent of an angle θ in standard position is the y-coordinate of the point where the terminal side of the angle intersects the tangent line x = 1 . x equals 1 .  A tangent function in the form y = a tan b θ y equals eh tangent b theta  has a period of π b . pi over b , .

Example

What is the period of y = tan π 4 θ ? y equals tangent , pi over 4 , theta . question mark  Tell where two asymptotes occur.

period = π b = π π 4 = 4 period , equals , pi over b , equals , fraction pi , over fraction pi , over 4 end fraction end fraction , equals 4

One cycle occurs in the interval from − 2 negative 2  to 2, so there are asymptotes at θ = − 2  and  θ = 2 . theta equals negative 2 , and , theta equals 2 .

Exercises

Graph each function in the interval from 0 to 2π. Then evaluate the function at t = π 4 t equals , pi over 4  and t = π 2 . t equals , pi over 2 . .  If the tangent is undefined at that point, write undefined.

28. y = tan 1 2 t y equals tangent , 1 half , t
29. y = tan 3 t y equals tangent 3 t
30. y = 2 tan t y equals 2 tangent t
31. y = 4 tan 2 t y equals 4 tangent 2 t

Table of Contents