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Exercises Trigonometry

Example 6 from Algebra and Trigonometry, 7.3 Unit Circle

Using Reference Angles to Find Sine and Cosine

ⓐ Using a reference angle, find the exact value of $150°$ and $sin (150°) .$
ⓑ Using the reference angle, find $\cos \frac{5 π}{4}$ and $\sin \frac{5 π}{4} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. ⓐ $150°$ is located in the second quadrant. The angle it makes with the x-axis is $180° - 150° = 30°,$ so the reference angle is $30° .$
  This tells us that $150°$ has the same sine and cosine values as $30°,$ except for the sign.
  
  $\begin{matrix} \cos (30°) = \frac{\sqrt{3}}{2} & and & \sin (30°) = \frac{1}{2} \end{matrix}$
  Since $150°$ is in the second quadrant, the x-coordinate of the point on the circle is negative, so the cosine value is negative. The y-coordinate is positive, so the sine value is positive.
  
  $\begin{matrix} \cos (150°) = - \frac{\sqrt{3}}{2} & and & \sin (150°) = \frac{1}{2} \end{matrix}$
2. ⓑ $\frac{5 π}{4}$ is in the third quadrant. Its reference angle is $\frac{5 π}{4} - π = \frac{π}{4} .$ The cosine and sine of $\frac{π}{4}$ are both $\frac{\sqrt{2}}{2} .$ In the third quadrant, both $x$ and $y$ are negative, so:
  $\begin{matrix} \cos \frac{5 π}{4} = - \frac{\sqrt{2}}{2} & and & \sin \frac{5 π}{4} = - \frac{\sqrt{2}}{2} \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

ⓐ Use the reference angle of $315°$ to find $\cos (315°)$ and $\sin (315°) .$
ⓑ Use the reference angle of $- \frac{π}{6}$ to find $\cos (- \frac{π}{6})$ and $\sin (- \frac{π}{6}) .$

How did it go?

PreviousFinding a Reference Angle Next Using the Unit Circle to Find Coordinates

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 7.3, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.