Question

Solving trigonometric equations: Give the two smallest solutions of cos(5theta)=0.1495, on [0,2pi). Where do I begin?

`cos(5theta)=-0.1495`
given: `[0,pi)`

I have no clue where to start here!!!

Answer 1

Please see below.

Explanation:

.

`cos(5theta)=-0.1495` Given `[0,pi)`

`arccos(-0.1495)=5theta`

`5theta=1.72 Radians=98.6^@`

`theta=1.42/5=0.344 Radians=19.72^@`

If what is listed in small print is correct and your domain is `[0, pi)` there is only one answer.

What you typed is `0.1495` but in small print it is `-0.1495` and you typed `[0,2pi)` but in small print it is `[0,pi)`

Answer 2

2 smallest:
`t = 16^@28; t = 55^@72`

Explanation:

cos 5t = 0.1495
Calculator and unit circle give 2 solutions for (5t):
`5t = +- 81^@40 + k360^@`

a. `5t = 81^@40 + k360^@`
`t = 16^@28 + k72^@`
k = 0 --> t = 16.28; k = 1 --> t = 88.28; k = 2 --> t = 160.28;
k = 3 --> t = 232.28; k = 4 --> t = 304.28

b. `5t = - 81^@40`, or `5t = 278^@60 + k360^@` (co-terminal)
`t = 55^@72 + k72^@`
k =0 --> t = 55.72; k = 1 --> t = 127.72; k = 2 --> t = 199.72;
k = 3 --> t = 271.72; k = 4 --> t = 343.72.
Therefor, for (0, 360), the 2 smallest values of t are:
`t = 16^@28`, and `t = 55^@72`