13-7 Translating Sine and Cosine Functions

Quick Review

Each horizontal translation of certain periodic functions is a phase shift. When g ( x ) = f ( x − h ) + k , g open x close equals f open x minus h close plus k comma  the value of h is the amount of the horizontal shift and the value of k is the amount of the vertical shift.

Example

What is an equation for the translation of y = sin x , 2 y equals sine x comma 2  units to the right and 1 unit up?

2 units to the right means h = 2 h equals 2  and 1 unit up means k = 1 . k equals 1 .

An equation is y = sin ( x − 2 ) + 1 . y equals sine open x minus 2 close plus 1 .

Exercises

Graph each function in the inteval from 0 to 2π.

32. y = cos ( x + π 2 ) y equals cosine . open . x plus , pi over 2 . close
33. y = 2 sin x − 4 y equals 2 sine x minus 4
34. y = sin ( x − π ) + 3 y equals sine open x minus pi close plus 3
35. y = cos ( x + π ) − 1 y equals cosine open x plus pi close minus 1

Write an equation for each translation.

36. y = sin x ,  π 4 y equals sine x comma , pi over 4  units to the right
37. y = cos x , y equals cosine x comma  2 units down

13-8 Reciprocal Trigonometric Functions

Quick Review

The cosecant (csc), secant (sec), and cotangent (cot) functions are defined as reciprocals for all real numbers θ (except those that make a denominator zero).

csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ co-secant theta equals . fraction 1 , over sine theta end fraction secant theta equals . fraction 1 , over cosine theta end fraction co-tangent theta equals . fraction 1 , over tangent theta end fraction

Example

Suppose sin θ = − 3 5 . sine theta equals negative , 3 fifths , .  Find csc θ.

csc θ = 1 sin θ = 1 − 3 5 = − 5 3 co-secant theta equals . fraction 1 , over sine theta end fraction . equals . fraction 1 , over fraction negative 3 , over 5 end fraction end fraction . equals negative , 5 thirds

Exercises

Evaluate each expression. Write your answer in exact form.

38. sec ( − 45 ° ) secant open negative 45 degrees close
39. cot 120 ° co-tangent , 120 degrees
40. csc 150 ° co-secant , 150 degrees
41. cot ( − 150 ° ) co-tangent open negative 150 degrees close

Graph each function in the interval from 0 to 4π.

42. y = 2 csc θ y equals 2 csc theta
43. y = sec θ − 1 y equals secant theta negative 1
44. y = cot 1 4 θ y equals co-tangent , 1 fourth , theta
45. y = csc 1 2 θ + 2 y equals co-secant , 1 half , theta plus 2

Table of Contents