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!!!

2 Answers
Jun 21, 2018

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)

Jun 21, 2018

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