How to Evaluate Sine on the Unit Circle

Step-by-step solution

1. Convert the angle to a point on the unit circle.

   $$45^\circ \Rightarrow \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$

   Why: A 45-degree angle cuts the first quadrant exactly in half, so the two coordinates match.

2. Remember which coordinate gives sine.

   $$\sin\theta = y$$

   Why: On the unit circle, the y-coordinate is the sine and the x-coordinate is the cosine.

3. Read the answer.

   $$\sin 45^\circ = \frac{\sqrt{2}}{2}$$

   Why: The y-coordinate of the point is sqrt(2)/2.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

1. 1

   $\sin 60^\circ$

   Show answer

   $\frac{\sqrt{3}}{2}$

2. 2

   $\sin 90^\circ$

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   $1$

3. 3

   $\sin 0^\circ$

   Show answer

   $0$

4. 4

   $\sin 30^\circ$

   Show answer

   $\frac{1}{2}$

5. 5

   $\sin 180^\circ$

   Show answer

   $0$