Mathematics › 🔍 Grade 10 › Unit Circle Radar: Sine and Cosine as Projections

Unit Circle Radar: Sine and Cosine as Projections

Sweep a radar arm around a circle of radius 1, and its shadow on each axis is exactly sine and cosine.

Grade 10Geometry / Algebra 2

Unit Circle Radar: Sine and Cosine as Projections — illustration

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The big idea: On a circle of radius 1 centered at the origin, any angle θ measured from the positive x-axis lands a point whose coordinates are simply (cos θ, sin θ). This frees sine and cosine from right triangles — they now make sense for any angle at all, in every direction, not just between 0° and 90°.

🎯 By the end, you'll be able to

Define cosine and sine as the x- and y-coordinates of a point on the unit circle
Extend right-triangle trigonometry beyond 0°–90° using the unit circle
Identify the sign of sine and cosine in each of the four quadrants
Find exact sine and cosine values for angles using reference angles

📎 You should already know

Right-triangle trigonometry (SOH-CAH-TOA)
Coordinate plane basics

Trigonometry outgrows the triangle

SOH-CAH-TOA works beautifully for a right triangle, but a right triangle only has angles between 0° and 90°. What is sin(150°)? Or cos(−40°)? There is no right triangle with an angle of 150° in it — so trigonometry needs a new home. That home is the unit circle: a circle of radius 1 centered at the origin.

🔑 Cosine and sine, redefined

Sweep a ray from the positive x-axis through angle θ. Where that ray meets the unit circle, the point's coordinates are the trig values: the x-coordinate is cos θ and the y-coordinate is sin θ.

\[ \cos^2\theta + \sin^2\theta = 1 \]

Because (cos θ, sin θ) is always a point on a circle of radius 1, its coordinates satisfy x² + y² = 1 for every angle θ.

🎮 Unit Circle Radar LIVE

Sweep the angle and read sine and cosine as the y- and x-projections of the point.

The four quadrants

As the radar arm sweeps past 90°, the point's x-coordinate goes negative while y stays positive — so cosine becomes negative while sine stays positive. Track the signs all the way around: Quadrant I has both positive, Quadrant II has cosine negative and sine positive, Quadrant III has both negative, and Quadrant IV has cosine positive and sine negative.

✨ Reference angles do the heavy lifting

Any angle can be tied back to an acute reference angle — its distance from the nearest x-axis. Find the ordinary right-triangle value for that reference angle, then attach the correct sign for the quadrant you're in.

📝 Worked example: Find cos(150°) and sin(150°).

150° is in Quadrant II, 30° short of 180° — so its reference angle is 30°.
In Quadrant II, cosine is negative and sine is positive.
cos(150°) = −cos(30°) = −√3⁄2. sin(150°) = sin(30°) = ½.

✓ cos(150°) = −√3&frasl;2, sin(150°) = ½.

📝 Worked example: Find cos(270°) and sin(270°).

270° points straight down the negative y-axis, so the point on the unit circle is (0, −1).
The x-coordinate is cos(270°) and the y-coordinate is sin(270°).

✓ cos(270°) = 0, sin(270°) = −1.

⚠️ Don't guess the sign — check the quadrant

It is easy to compute the reference-angle value correctly and then attach the wrong sign. Always check which quadrant the angle actually lands in before deciding whether cosine and sine should be positive or negative there.

Check your understanding

1. On the unit circle, the point where the terminal ray of angle θ meets the circle has coordinates:

By definition on the unit circle, the x-coordinate of that point is cos θ and the y-coordinate is sin θ.

2. In which quadrant are both sine and cosine negative?

Quadrant III (between 180° and 270°) has both x and y coordinates negative, so both cosine and sine are negative there.

3. What is cos(150°)?

150° has reference angle 30° and lies in Quadrant II, where cosine is negative: cos(150°) = −cos(30°) = −√3/2.

4. What is sin(270°)?

270° lands at the point (0, −1) on the unit circle, so sin(270°), its y-coordinate, is −1.

5. Why does cos²θ + sin²θ = 1 hold true for every angle θ?

Every point (cos θ, sin θ) sits on the unit circle, whose equation is x² + y² = 1 — that's exactly cos²θ + sin²θ = 1.

✅ Key takeaways

The unit circle defines cos θ and sin θ as the x- and y-coordinates of a point at angle θ from the positive x-axis.
This extends sine and cosine to every angle, not just the 0°–90° range a right triangle allows.
The sign of cosine and sine depends on the quadrant: (+,+), (−,+), (−,−), (+,−) going around.
A reference angle ties any angle back to an acute angle whose trig values you already know.
Because every point on the unit circle satisfies x² + y² = 1, cos²θ + sin²θ = 1 for every angle.