  ### Home > INT3 > Chapter 12 > Lesson 12.1.3 > Problem12-45

12-45.

For each of the following equations, list all of the solutions without using a calculator. Draw a graph or unit circle to support your answers. Then predict the solution that your calculator will give, and use your calculator to check your prediction. Homework Help ✎

1. $4 \operatorname { sin } ( x ) + 2 = 0$

Subtract $2$ from both sides and divide by $4$ to solve for $\sin(x)$.

$\sin(x)=1/2$

On a unit circle, find the two different times $\sin(x)=1/2$. To give all of the solutions, remember that every time you go around the circle, that value occurs again. Since there are $2π$ radians in a circle, add $2πn$ to each solution.

$x = \frac{7\pi}{6}+2{\pi}n,\;\frac{11\pi}{6}+2{\pi}n$

1. $2\cos(x)=\sqrt{3}$

Use the same method as part (a).

$x = \frac{\pi}{6}+2{\pi}n,\;\frac{11\pi}{6}+2{\pi}n$

1. $\operatorname { tan } ( x ) + 1 = 0$

Use the same method as part (a). How is the tangent function different from the sine function?

1. $4 \operatorname { cos } ( x ) - 4 = 0$

$x=2\pi{n}$ 