The Unit Circle: Sine and Cosine Defined Geometrically

The unit circle is the geometric foundation of all trigonometry. A circle of radius 1 centered at the origin gives sine and cosine a beautiful, intuitive definition: as a point moves around the circle, its $x$-coordinate is $\cos\theta$ and its $y$-coordinate is $\sin\theta$.

This single picture explains why sine and cosine oscillate between $-1$ and $1$, why they're 90° out of phase, and why the identity $\sin^{2}\theta + \cos^{2}\theta = 1$ holds for every angle.

The Definition

For a point on the unit circle at angle $\theta$ (measured counterclockwise from the positive $x$-axis):

$P = (\cos\theta,\, \sin\theta)$

Because the point lies on a circle of radius 1, the distance from the origin is exactly 1:

$\cos^{2}\theta + \sin^{2}\theta = 1 \quad\text{(the Pythagorean identity)}$

Step-by-Step Solution

Given: A point $P$ moving around the unit circle, parameterized by angle $\theta$ from the positive $x$-axis.

Find: The exact $(x, y)$ coordinates at the famous angles 0, 30°, 45°, 60°, and 90°.

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Step 1 — At $\theta = 0°$.

$P = (\cos 0°,\, \sin 0°) = (1, 0)$

The point sits on the positive $x$-axis.

Step 2 — At $\theta = 30°$ ($= \pi/6$).

Using a 30°-60°-90° right triangle inscribed in the circle:

$\cos 30° = \dfrac{\sqrt{3}}{2} \approx 0.866 \qquad \sin 30° = \dfrac{1}{2} = 0.500$

$P = \left(\dfrac{\sqrt{3}}{2},\; \dfrac{1}{2}\right) \approx (0.866,\, 0.500)$

Step 3 — At $\theta = 45°$ ($= \pi/4$).

Using a 45°-45°-90° isoceles right triangle:

$\cos 45° = \sin 45° = \dfrac{\sqrt{2}}{2} \approx 0.707$

$P = \left(\dfrac{\sqrt{2}}{2},\; \dfrac{\sqrt{2}}{2}\right) \approx (0.707,\, 0.707)$

Step 4 — At $\theta = 60°$ ($= \pi/3$).

By the symmetry of the 30°-60°-90° triangle, sine and cosine swap their roles:

$\cos 60° = \dfrac{1}{2} \qquad \sin 60° = \dfrac{\sqrt{3}}{2}$

$P = \left(\dfrac{1}{2},\; \dfrac{\sqrt{3}}{2}\right) \approx (0.500,\, 0.866)$

Step 5 — At $\theta = 90°$ ($= \pi/2$).

The point is now straight up:

$P = (\cos 90°,\, \sin 90°) = (0, 1)$

Step 6 — Verify the Pythagorean identity at $\theta = 30°$.

$\cos^{2} 30° + \sin^{2} 30° = \left(\dfrac{\sqrt{3}}{2}\right)^{2} + \left(\dfrac{1}{2}\right)^{2} = \dfrac{3}{4} + \dfrac{1}{4} = 1 \;\;\checkmark$

The identity holds at every angle by the very definition of the unit circle.

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Answer: As $\theta$ moves around the unit circle, the point $P$ traces out the position $(\cos\theta,\, \sin\theta)$. The famous values are:

• $\theta = 0°$: $(1, 0)$
• $\theta = 30°$: $(\sqrt{3}/2,\, 1/2)$
• $\theta = 45°$: $(\sqrt{2}/2,\, \sqrt{2}/2)$
• $\theta = 60°$: $(1/2,\, \sqrt{3}/2)$
• $\theta = 90°$: $(0, 1)$

The Pythagorean identity $\cos^{2}\theta + \sin^{2}\theta = 1$ holds for every $\theta$ — that's just saying the point stays on a circle of radius 1.

Try It

• Slide the angle widget — watch the point move around the circle.
• The horizontal projection to the $x$-axis traces out $\cos\theta$.
• The vertical projection to the $y$-axis traces out $\sin\theta$.
• The HUD shows live values of both, updated in real time.