# How do you find all solutions to sin (2x + 1) = 0.2?

It has infinite set of solutions

#### Explanation:

The general solution is given as follows
$\setminus \sin \left(2 x + 1\right) = 0.2$
$2 x + 1 = 2 n \setminus \pi + \setminus {\sin}^{- 1} \left(0.2\right)$ OR $2 x + 1 = 2 n \setminus \pi + \setminus \pi - \setminus {\sin}^{- 1} \left(0.2\right)$
$x = \setminus \frac{2 n \setminus \pi - 1}{2} + \setminus \frac{1}{2} \setminus {\sin}^{- 1} \left(0.2\right)$ OR $x = \setminus \frac{\left(2 n + 1\right) \setminus \pi - 1}{2} - \setminus \frac{1}{2} \setminus {\sin}^{- 1} \left(0.2\right)$
where, $n$ is any integer i.e. $n = 0 , \setminus \pm 1 , \setminus \pm 2 , \setminus \pm 3 , \setminus \ldots$

Jun 19, 2018

$x = {55}^{\circ} 57 + k {360}^{\circ}$
$x = {337}^{\circ} 11 + k {360}^{\circ}$

#### Explanation:

1 is expressed in radians. We can convert it to degrees for easier solving.
$\pi = 3.14$ --> ${180}^{\circ}$
1 radian --> $\frac{180}{3.14} = {57}^{\circ} 32$
sin (2x + 57.32) = 0.2
Calculator and unit circle give 2 solutions for (2x + 57.32):
a. $\left(2 x + 57.32\right) = {11}^{\circ} 54$
$2 x = 11.54 - 57.32 = - {45}^{\circ} 78$
$x = - {22}^{\circ} 89$, or $x = 360 - 22.89 = {337}^{\circ} 11$ (co-terminal).
b. $\left(2 x + 57.32\right) = 180 - 11.54 = 168.46$
$2 x = 168.46 - 57.32 = 111.14$
$x = {55}^{\circ} 57$
For general answers, add $k {360}^{\circ}$
Check by calculator.
$x = 55.57 - \to 2 x = 111.14 - \to \sin \left(2 x + 57.32\right) = \sin \left(168.46\right) = 0.20$. Proved