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9-78.

Calculate the exact values of the angles that are solutions to the equation $\text{sin}\left(θ\right) = 0.5$. Express your solutions in radians. 9-78 HW eTool (Desmos)  Homework Help ✎

Draw a unit circle and a triangle with the opposite side of $0.5$ and hypotenuse $1$.
Since the short leg is half the hypotenuse, this is a $30^\circ-60^\circ-90^\circ$ triangle. $θ = 30^\circ$.

Reflect the triangle across the $y$-axis. If $30^\circ$ is the reference angle, what is the circular angle?
Notice that $θ = 150^\circ$ as well.

Convert both angles to radians. Recall that $180^\circ$ is $π$ radians. What fraction of $180^\circ$ is $150^\circ$?

Every time you go around the circle, you will come back to these points, so there are an infinite number of solutions to this equation.
A new solution occurs every $360^\circ$ or $2π$ radians.

$\frac{\pi}{6} \pm 2\pi, \frac{5 \pi}{6} \pm 2 \pi$

Use the eTool below to view the solution as a graph.
Click the link at right for the full version of the eTool: INT3 9-78 HW eTool.