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Using the Unit Circle to Find Coordinates · Physics Playground

Exercises Trigonometry

Example 7 from Algebra and Trigonometry, 7.3 Unit Circle

Using the Unit Circle to Find Coordinates

Find the coordinates of the point on the unit circle at an angle of $\frac{7 π}{6} .$

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.

Solution
We know that the angle $\frac{7 π}{6}$ is in the third quadrant.

First, let’s find the reference angle by measuring the angle to the x-axis. To find the reference angle of an angle whose terminal side is in quadrant III, we find the difference of the angle and $π .$

$\frac{7 π}{6} - π = \frac{π}{6}$

Next, we will find the cosine and sine of the reference angle.

$\begin{matrix} \cos (\frac{π}{6}) = \frac{\sqrt{3}}{2} & \sin (\frac{π}{6}) = \frac{1}{2} \end{matrix}$

We must determine the appropriate signs for x and y in the given quadrant. Because our original angle is in the third quadrant, where both $x$ and $y$ are negative, both cosine and sine are negative.

$\begin{matrix} \cos (\frac{7 π}{6}) & = & - \frac{\sqrt{3}}{2} \\ \sin (\frac{7 π}{6}) & = & - \frac{1}{2} \end{matrix}$
Now we can calculate the $(x, y)$ coordinates using the identities $x = \cos θ$ and $y = \sin θ .$
The coordinates of the point are $(- \frac{\sqrt{3}}{2}, - \frac{1}{2})$ on the unit circle.

Your turn

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

Find the coordinates of the point on the unit circle at an angle of $\frac{5 π}{3} .$

How did it go?

PreviousUsing Reference Angles to Find Sine and Cosine Next Finding Trigonometric Functions from a Point on the Unit Circle

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.