How to Solve: Unit Circle Basics

How to Solve: Unit Circle Basics

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Master the unit circle, and you’ll unlock every trigonometry problem—from SATs to AP Calculus—where angles, sine, cosine, or radians appear. No more guessing!

What You Need To Know First

Before diving into the unit circle, ensure you understand: 1. Right-triangle trigonometry (SOH-CAH-TOA: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent). 2. Degrees vs. radians (180° = π radians; 360° = 2π radians). 3. Special right triangles (30-60-90 and 45-45-90 triangles and their side ratios).

Key Vocabulary

Formulas To Know

Step-by-Step Method

Goal: Find the sine, cosine, or tangent of any angle using the unit circle.

Step 1: Identify the Angle’s Quadrant

• 0° to 90° (0 to π/2): Quadrant I (all trig values positive).
• 90° to 180° (π/2 to π): Quadrant II (sine positive; cosine/tangent negative).
• 180° to 270° (π to 3π/2): Quadrant III (tangent positive; sine/cosine negative).
• 270° to 360° (3π/2 to 2π): Quadrant IV (cosine positive; sine/tangent negative).

Why? Signs of sine/cosine depend on the quadrant.

Step 2: Find the Reference Angle

• Quadrant I: Reference angle = angle itself.
• Quadrant II: Reference angle = 180° – angle (or π – angle in radians).
• Quadrant III: Reference angle = angle – 180° (or angle – π).
• Quadrant IV: Reference angle = 360° – angle (or 2π – angle).

Why? The reference angle helps you use the 30-60-90 or 45-45-90 triangle ratios.

Step 3: Recall the Coordinates for Key Angles

Memorize these exact values for 0°, 30°, 45°, 60°, and 90° (and their radian equivalents):

Why? These are the "building blocks" for all other angles.

Step 4: Determine the Signs Based on Quadrant

• Quadrant I: (cos θ, sin θ) = (+, +)
• Quadrant II: (cos θ, sin θ) = (–, +)
• Quadrant III: (cos θ, sin θ) = (–, –)
• Quadrant IV: (cos θ, sin θ) = (+, –)

Why? The signs change based on the quadrant, but the reference angle’s values stay the same.

Step 5: Write the Final Coordinates

Combine the reference angle’s coordinates with the correct signs for the quadrant.

Step 6: Find Sine, Cosine, or Tangent

• sin θ = y-coordinate
• cos θ = x-coordinate
• tan θ = y/x (or sin θ / cos θ)

Worked Examples

Example 1 – Basic: Find sin(150°) and cos(150°)

Step 1: 150° is in Quadrant II (90° < 150° < 180°). Step 2: Reference angle = 180° – 150° = 30°. Step 3: For 30°, (cos, sin) = (√3/2, 1/2). Step 4: In Quadrant II, cosine is negative, sine is positive. Step 5: Coordinates = (–√3/2, 1/2). Step 6: - sin(150°) = 1/2 - cos(150°) = –√3/2

What we did and why: We used the reference angle (30°) to find the base values, then adjusted the signs for Quadrant II.

Example 2 – Medium: Find tan(5π/4)

Step 1: 5π/4 radians = 225° (since π = 180°). This is in Quadrant III (180° < 225° < 270°). Step 2: Reference angle = 225° – 180° = 45° (or 5π/4 – π = π/4). Step 3: For 45°, (cos, sin) = (√2/2, √2/2). Step 4: In Quadrant III, both cosine and sine are negative. Step 5: Coordinates = (–√2/2, –√2/2). Step 6: - tan(5π/4) = sin/cos = (–√2/2) / (–√2/2) = 1

What we did and why: We converted radians to degrees for clarity, found the reference angle, and used the Quadrant III signs to get the correct tangent value.

Example 3 – Exam Style: If cos θ = –3/5 and θ is in Quadrant II, find sin θ.

Step 1: Given cos θ = –3/5, and θ is in Quadrant II. Step 2: On the unit circle, cos θ = x-coordinate = –3/5. Step 3: Use the unit circle equation: x² + y² = 1. - (–3/5)² + y² = 1 - 9/25 + y² = 1 - y² = 1 – 9/25 = 16/25 - y = ±4/5 Step 4: In Quadrant II, sine (y) is positive. Step 5: sin θ = 4/5

What we did and why: We used the unit circle equation to find the missing y-coordinate (sine) and applied the Quadrant II sign rule.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in. The unit circle is just a circle with radius 1. Every point on it is (cos θ, sin θ). Memorize the key angles: 0°, 30°, 45°, 60°, 90°—and their radian versions. For any angle, find its quadrant, then its reference angle. Use the reference angle’s values, but adjust the signs based on the quadrant. Quadrant I: all positive. Quadrant II: sine positive. Quadrant III: tangent positive. Quadrant IV: cosine positive. If you forget, draw the circle and label the signs. For tangent, just divide sine by cosine. And if the angle is negative or over 360°, add or subtract 360° to find a coterminal angle. That’s it—you’ve got this!