All courses Algebra II · F-TF.2 41 of 55
Use the unit circle to extend trig functions to all real-number radian measures.

Find cos(θ) and sin(θ) from a point on the unit circle

Problem
If a point on the unit circle is \((1/2,\sqrt{3}/2)\), what are \(\cos(\theta)\) and \(\sin(\theta)\)?
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Hint

On the unit circle, the x-coordinate is cos(theta) and the y-coordinate is sin(theta).

Match the first coordinate to cosine and the second coordinate to sine; do not swap the coordinate order.

Solution walkthrough

01

Use the unit-circle coordinate definition

\[(x,y)=(\cos(\theta),\sin(\theta))\]

A terminal point on the unit circle records cosine as its x-coordinate and sine as its y-coordinate.

02

Read the two coordinates

\[x=1/2~\text{and}~y=\sqrt{3}/2\]

The given terminal point is (1/2,sqrt(3)/2), so these are the cosine and sine values respectively.

03

Check that the point is on the unit circle

\[x^2+y^2=(1/2)^2+(\sqrt{3}/2)^2=1/4+3/4=1\]

The coordinates satisfy the unit-circle equation, confirming that the values are consistent.

04

State cosine and sine

\[\cos(\theta)~=~1/2,~\sin(\theta)~=~\sqrt{3}/2\]

Cosine is the horizontal coordinate and sine is the vertical coordinate of the terminal point.

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Another way

  1. Recognize the point as the standard pi/3 unit-circle point and recall cos(pi/3)=1/2 and sin(pi/3)=sqrt(3)/2.

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Common mistake

Do not reverse the coordinates. On the unit circle, cosine is x and sine is y.

Answer diagram for M3-034-A01-V01: the same geometry with cosine and sine identified as the x- and y-coordinates.

Solution walkthrough video