Precalc
Section 5.2

Unit Circle — Sine & Cosine

The unit circle is the foundation of everything in trig. You'll use it directly in Sections 5.3–5.4, it's behind every trig graph in Chapter 6, and you need fast unit-circle lookups to solve equations in Section 7.5. Expect it on almost every Exam 3 problem.

⚡ Quick Summary

Picture a clock hand starting at 3 o'clock (pointing right) and sweeping counterclockwise around a circle with radius 1. At any position, the hand's tip lands on a point — and the cosine of the angle is just how far that point is to the right or left (the xx-coordinate), while the sine is how far up or down it is (the yy-coordinate). That's the entire unit circle idea: trig values are just coordinates on a circle.

Unit circle definitions
cos⁡θ=x,sin⁡θ=y\cos\theta = x, \quad \sin\theta = y

where (x,y)(x, y) is the point on the unit circle at angle θ\theta

Pythagorean identity
cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1

follows directly from x2+y2=1x^2 + y^2 = 1

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

Picture a clock hand starting at 3 o'clock (pointing right) and sweeping counterclockwise around a circle with radius 1. At any position, the hand's tip lands on a point — and the cosine of the angle is just how far that point is to the right or left (the xx-coordinate), while the sine is how far up or down it is (the yy-coordinate). That's the entire unit circle idea: trig values are just coordinates on a circle.

You only need to memorize five angles in the first quadrant (the upper-right quarter): 0°,30°,45°,60°,90°0°, 30°, 45°, 60°, 90°. Their sine values follow a neat pattern: 0,12,22,32,10, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1. Cosine uses the exact same values but in reverse order. Once you know these five, you can find *any* trig value on the entire circle.

For angles outside the first quadrant, find the reference angle — that's the shortcut back to one of those five angles you already know. The reference angle is the acute angle between your angle's endpoint and the nearest part of the xx-axis. You use the reference angle to get the *size* of the trig value, then just pick the right sign (positive or negative) based on which quadrant you're in.

The mnemonic All Students Take Calculus tells you which trig functions are positive in each quadrant: All (QI — everything positive), Sine (QII — only sine positive), Tangent (QIII — only tangent positive), Cosine (QIV — only cosine positive). This makes sense if you think about the coordinates: in QII the point is to the left (xx negative) but above (yy positive), so cosine is negative but sine is positive.

When a problem gives you a trig value and asks for the angle, work backwards: find the reference angle from the positive version of the value, then place it in every quadrant where the function has the given sign. For cos⁡θ=−32\cos\theta = -\frac{\sqrt{3}}{2}, the reference angle is π6\frac{\pi}{6} (since cos⁡π6=32\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}), and cosine is negative in QII and QIII, giving you two answers.

Interactive Diagram

Key Formulas

Unit circle definitions
cos⁡θ=x,sin⁡θ=y\cos\theta = x, \quad \sin\theta = y

where (x,y)(x, y) is the point on the unit circle at angle θ\theta

Pythagorean identity
cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1

follows directly from x2+y2=1x^2 + y^2 = 1

Key first-quadrant values
sin⁡0=0,  sin⁡π6=12,  sin⁡π4=22,  sin⁡π3=32,  sin⁡π2=1\sin 0 = 0,\; \sin\frac{\pi}{6} = \frac{1}{2},\; \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2},\; \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2},\; \sin\frac{\pi}{2} = 1
Reference angle
ref(θ)=acute angle between terminal side and x-axis\text{ref}(\theta) = \text{acute angle between terminal side and } x\text{-axis}

use the reference angle to find trig values in any quadrant

Even/odd identities
cos⁡(−θ)=cos⁡θ,sin⁡(−θ)=−sin⁡θ\cos(-\theta) = \cos\theta, \quad \sin(-\theta) = -\sin\theta

cosine is even, sine is odd

Key Takeaways

  • ✓On the unit circle: cos⁡θ=x\cos\theta = x-coordinate, sin⁡θ=y\sin\theta = y-coordinate
  • ✓Sine values for key angles follow the pattern 02,12,22,32,42\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}; cosine is the reverse order
  • ✓All Students Take Calculus → QI all positive, QII sine positive, QIII tangent positive, QIV cosine positive
  • ✓Reference angle is always measured to the xx-axis: QII subtract from π\pi, QIII subtract π\pi, QIV subtract from 2π2\pi

⚠️ Common Mistakes

  • ✗Measuring the reference angle to the yy-axis instead of the xx-axis — it's always the acute angle to the nearest part of the xx-axis
  • ✗Getting the ASTC signs wrong: cosine is the xx-coordinate (negative on the left side), sine is the yy-coordinate (negative below)
  • ✗Confusing sin⁡π3\sin\frac{\pi}{3} and sin⁡π6\sin\frac{\pi}{6}: sin⁡π3=32\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} (bigger angle, bigger sine) and sin⁡π6=12\sin\frac{\pi}{6} = \frac{1}{2}
  • ✗Forgetting the even/odd identities: cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos\theta (even) but sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin\theta (odd)
Next: Section 5.3Builds on: 5.1
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5.1 Angles & Radian Measure
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5.3 The Other Trigonometric Functions