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

## $\cos \left(5 \theta\right) = - 0.1495$ given: $\left[0 , \pi\right)$ I have no clue where to start here!!!

Jun 21, 2018

#### Explanation:

.

$\cos \left(5 \theta\right) = - 0.1495$ Given $\left[0 , \pi\right)$

$\arccos \left(- 0.1495\right) = 5 \theta$

$5 \theta = 1.72 R a \mathrm{di} a n s = {98.6}^{\circ}$

$\theta = \frac{1.42}{5} = 0.344 R a \mathrm{di} a n s = {19.72}^{\circ}$

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

What you typed is $0.1495$ but in small print it is $- 0.1495$ and you typed $\left[0 , 2 \pi\right)$ but in small print it is $\left[0 , \pi\right)$

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):
$5 t = \pm {81}^{\circ} 40 + k {360}^{\circ}$

a. $5 t = {81}^{\circ} 40 + k {360}^{\circ}$
$t = {16}^{\circ} 28 + k {72}^{\circ}$
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. $5 t = - {81}^{\circ} 40$, or $5 t = {278}^{\circ} 60 + k {360}^{\circ}$ (co-terminal)
$t = {55}^{\circ} 72 + k {72}^{\circ}$
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:
$t = {16}^{\circ} 28$, and $t = {55}^{\circ} 72$