5.2Sine and Cosine on the Unit Circle
Define sin θ and cos θ as the y- and x-coordinates on the unit circle. Evaluate exact values at 0, π/6, π/4, π/3, π/2, and their multiples. Apply the Pythagorean identity sin²θ + cos²θ = 1.
The unit circle is the foundation of all trigonometry. Memorizing exact values at the 16 key angles lets you evaluate trig functions instantly — a skill required for every calculus problem involving trig.
Essential Question
How do we define sine and cosine for any angle — not just acute angles in a right triangle?
Lesson Overview
The unit circle is a circle of radius 1 centered at the origin. For any angle θ measured counterclockwise from the positive x-axis, the terminal side of the angle intersects the unit circle at exactly one point. We define cosine and sine using that intersection point: the x-coordinate is cos θ and the y-coordinate is sin θ. This extends trig beyond right triangles to all real-number inputs.
The Unit Circle — 16 Key Points
Each point is labeled with its angle in radians and its (cos θ, sin θ) coordinate pair. Memorize the first quadrant values — the rest follow by symmetry.
Key Vocabulary
Unit Circle
A circle with radius 1 centered at the origin. Equation: x² + y² = 1.
Example: The point (√2/2, √2/2) lies on the unit circle because (√2/2)² + (√2/2)² = 1.
Cosine (cos θ)
The x-coordinate of the point where the terminal side of angle θ meets the unit circle.
Example: cos(π/3) = 1/2 because the point at π/3 is (1/2, √3/2).
Sine (sin θ)
The y-coordinate of the point where the terminal side of angle θ meets the unit circle.
Example: sin(π/3) = √3/2 because the point at π/3 is (1/2, √3/2).
Reference Angle
The acute angle formed between the terminal side of θ and the x-axis. Always between 0 and π/2.
Example: The reference angle for 5π/6 is π/6, since 5π/6 is in Q2 and π − 5π/6 = π/6.
Pythagorean Identity
sin²θ + cos²θ = 1 for all angles θ. Follows directly from x² + y² = 1 on the unit circle.
Signs by Quadrant — ASTC
Remember "All Students Take Calculus": the first letter of each word tells you which function is positive in that quadrant (Q1 → Q4).
Using Reference Angles
To evaluate sin or cos at any angle: (1) find the reference angle, (2) evaluate at that acute angle, (3) apply the correct sign for the quadrant.
Pythagorean Identity
Because every point on the unit circle satisfies x² + y² = 1, substituting x = cos θ and y = sin θ gives sin²θ + cos²θ = 1 for every angle θ.
Even and Odd Properties
Cosine is EVEN
cos(−θ) = cos θ
The x-coordinate is unchanged by reflection across the x-axis.
Sine is ODD
sin(−θ) = −sin θ
The y-coordinate is negated by reflection across the x-axis.
Period of Sine and Cosine
Both sin and cos have period 2π:
sin(θ + 2π) = sin θ
cos(θ + 2π) = cos θ
Going around the full circle (2π radians) returns you to the same point, so the values repeat.
Worked Examples
Find the exact values of sin(π/4) and cos(π/4).
The angle π/4 is in Q1, so both sin and cos are positive.
The key point at π/4 on the unit circle is (√2/2, √2/2).
cos(π/4) = x-coordinate = √2/2
sin(π/4) = y-coordinate = √2/2
Find sin(5π/6) and cos(5π/6) using a reference angle.
5π/6 is in Q2 (between π/2 and π).
Reference angle: π − 5π/6 = π/6.
sin(π/6) = 1/2, cos(π/6) = √3/2.
In Q2: sin is positive, cos is negative.
So sin(5π/6) = +1/2, cos(5π/6) = −√3/2.
If sin θ = 3/5 and θ is in Q2, find cos θ.
Use the Pythagorean identity: sin²θ + cos²θ = 1.
(3/5)² + cos²θ = 1
9/25 + cos²θ = 1
cos²θ = 1 − 9/25 = 16/25
cos θ = ±4/5. Since θ is in Q2, cos θ is negative.
Evaluate cos(−π/3).
Cosine is an even function: cos(−θ) = cos θ.
cos(−π/3) = cos(π/3).
The point at π/3 is (1/2, √3/2), so cos(π/3) = 1/2.
Find all angles θ in [0, 2π) where sin θ = −√3/2.
sin θ = −√3/2 is negative, so θ is in Q3 or Q4.
The reference angle: sin(π/3) = √3/2, so θ_ref = π/3.
Q3: θ = π + π/3 = 4π/3.
Q4: θ = 2π − π/3 = 5π/3.
Guided Practice
Find the exact values of sin(2π/3) and cos(2π/3).
Hint: 2π/3 is in Q2. The reference angle is π − 2π/3. Use the Q2 sign rules.
If cos θ = −5/13 and θ is in Q3, find sin θ.
Hint: Apply sin²θ + cos²θ = 1. In Q3, sin is also negative.
Evaluate sin(−π/6) using the odd-function property.
Hint: sin(−θ) = −sin θ. Then look up sin(π/6) from the unit circle.
Find all θ in [0, 2π) where cos θ = √2/2.
Hint: cos is positive in Q1 and Q4. The reference angle has cos = √2/2.
Verify that the point (−√3/2, 1/2) lies on the unit circle and identify its angle.
Hint: Check x² + y² = 1. Then match the coordinates to a key angle in Q2.
Quick Check
Interactive Practice — 3 Questions
What is the y-coordinate of the point on the unit circle at angle θ = π/6?
In which quadrant is sin θ positive and cos θ negative?
Which identity is always true for any angle θ?
Common Mistakes
Common Mistakes
Writing sin(π/6) = √3/2 (confusing sin and cos at π/6)
sin(π/6) = 1/2 and cos(π/6) = √3/2. The smaller angle has the smaller sine.
Forgetting the sign when using a reference angle (e.g., writing cos(5π/6) = √3/2)
Always apply the quadrant sign rule after finding the reference-angle value. cos(5π/6) = −√3/2.
Thinking cos is odd: cos(−θ) = −cos θ
Cosine is EVEN: cos(−θ) = cos θ. Only sine is odd.
Solving sin²θ = 1 − cos²θ and forgetting the ± when taking the square root
cos θ = ±√(1 − sin²θ). Use the quadrant to choose the correct sign.
Math Tips
Math Tips
Memorize Q1 values only: (0,1), (π/6,1/2), (π/4,√2/2), (π/3,√3/2), (π/2,1). All other values follow by symmetry.
The "hand trick": hold up your left hand, fold down the finger for the angle (thumb=0, index=π/6, middle=π/4, ring=π/3, pinky=π/2). Count remaining fingers for the numerator under √.
Period means cos(θ + 2πk) = cos θ for any integer k. Use this to reduce large angles.
Reference angle formulas: Q2 → π − θ, Q3 → θ − π, Q4 → 2π − θ.
The Pythagorean identity has two useful rearrangements: sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ.