Unit Circle Math Example 1

Follow the full solution, then compare it with the other examples linked below.

easy

Verify that the point

\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)

lies on the unit circle and identify the angle

\theta

1
Check $x^2+y^2=1$ : $\left(\frac{\sqrt{3}}{2}\right)^2+\left(\frac{1}{2}\right)^2 = \frac{3}{4}+\frac{1}{4}=1$ . ✓ On the unit circle.
2
Identify angle: $\cos\theta=\frac{\sqrt{3}}{2}$ and $\sin\theta=\frac{1}{2}$ (both positive → first quadrant).
3
$\cos\theta=\frac{\sqrt{3}}{2}$ and $\sin\theta=\frac{1}{2}$ corresponds to $\theta=\frac{\pi}{6}$ (30°).

Answer

Point is on unit circle;

\theta = \dfrac{\pi}{6}

(30°)

Every point on the unit circle satisfies

x^2+y^2=1

, with

x=\cos\theta

and

y=\sin\theta

. Recognizing standard values of cosine and sine allows immediate identification of the corresponding angle.

About Unit Circle

The circle of radius 1 centered at the origin in the coordinate plane, used to define trigonometric functions for all angles.

Find [formula], [formula], and [formula] for [formula] using the unit circle. Identify which quadran

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