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Mom's Basement University

Trigonometric Functions on the Unit Circle

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll use the unit circle to quickly find the exact values of sine, cosine, and tangent for the most important angles, without a calculator.

The Concept

Section titled “The Concept”

On the unit circle, any point has coordinates (cos θ, sin θ). This means you can read off exact trig values just by knowing where the angle lands on the circle.

You only need to memorize the first quadrant values. The other three quadrants use the same numbers, just with different signs.

First Quadrant (0 to π/2): all trig functions are positive

  • 0 → (1, 0)
  • π/6 (30°) → (√3/2, 1/2)
  • π/4 (45°) → (√2/2, √2/2)
  • π/3 (60°) → (1/2, √3/2)
  • π/2 (90°) → (0, 1)

For the other quadrants, use the sign rules:

  • Q I: All positive
  • Q II: Sine positive, cosine and tangent negative
  • Q III: Tangent positive, sine and cosine negative
  • Q IV: Cosine positive, sine and tangent negative

A common mnemonic for this is ASTC (“All Students Take Calculus”): All, Sine, Tangent, Cosine - telling you which function is positive in each quadrant.

Reference angles are the key to using this. The reference angle is the acute angle between your terminal side and the x-axis. Find the reference angle, look up its first-quadrant values, then apply the correct sign for the quadrant you’re in.

Tangent is always sin θ / cos θ. If you know sin and cos, you know tan.

Worked Example

Section titled “Worked Example”

1. Find sin(2π/3) and cos(2π/3)

2π/3 = 120°, which is in Q II. The reference angle is π - 2π/3 = π/3.

In Q II, sine is positive and cosine is negative:

sin⁡(2π3)=+sin⁡(π3)=32\sin\left(\frac{2\pi}{3}\right) = +\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} cos⁡(2π3)=−cos⁡(π3)=−12\cos\left(\frac{2\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) = -\frac{1}{2}

2. Find tan(5π/4)

5π/4 = 225°, which is in Q III. The reference angle is 5π/4 - π = π/4.

In Q III, tangent is positive:

tan⁡(5π4)=+tan⁡(π4)=1\tan\left(\frac{5\pi}{4}\right) = +\tan\left(\frac{\pi}{4}\right) = 1

3. Find sin(7π/6)

7π/6 = 210°, which is in Q III. The reference angle is 7π/6 - π = π/6.

In Q III, sine is negative:

sin⁡(7π6)=−sin⁡(π6)=−12\sin\left(\frac{7\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2}

Real-World Application

Section titled “Real-World Application”

The unit circle is used in:

  • Computer graphics (rotating objects smoothly, calculating positions on circular paths)
  • Physics (circular motion, waves, pendulums)
  • Engineering (signal processing, AC circuits)
  • Music (sound wave analysis and synthesis)
  • Navigation and robotics (calculating headings and turns)

Example: when animating a spinning wheel or calculating the position of a planet in its orbit, programmers and engineers constantly use unit circle values.

Quiz

Section titled “Quiz”
On the unit circle, the coordinates of any point are:
What is $\cos\left(\frac{\pi}{3}\right)$?
What is $\sin\left(\frac{3\pi}{2}\right)$?
In the second quadrant, which trig function is positive?
The reference angle for $\frac{5\pi}{6}$ is: