Reference Angle Calculator

Find the reference angle for any angle in standard position with step-by-step solutions

Calculate Reference Angle

Any positive or negative angle is accepted

Reference Angle Results

Step 1: Normalize Angle

Original angle: 0°

Normalized angle: 0.0000°

Quadrant: 1

Step 2: Apply Formula

Formula: Reference angle = angle

Calculation: 0.0000°

0.0000°
Reference Angle (degrees)
0.0000 rad
Reference Angle (radians)

Quadrant 1 Properties

Trigonometric Function Signs:

sin θ: +

cos θ: +

tan θ: +

ASTC Rule:

All positive (All)

Common Reference Angles

30°
π/6 rad
45°
π/4 rad
60°
π/3 rad
90°
π/2 rad

Example Calculation

Example: Find reference angle for 250°

Step 1: Normalize the angle (250° is already between 0° and 360°)

Step 2: Determine quadrant (250° is in Quadrant III: 180° < θ < 270°)

Step 3: Apply formula for Quadrant III: Reference angle = angle - 180°

Step 4: Calculate: 250° - 180° = 70°

Result

Reference angle: 70° (or 1.2217 radians)

Trigonometric values: Use 70° values with appropriate signs for Quadrant III

Quadrant Guide

I

Quadrant I

0° to 90° (0 to π/2)

All functions positive

II

Quadrant II

90° to 180° (π/2 to π)

Only sine positive

III

Quadrant III

180° to 270° (π to 3π/2)

Only tangent positive

IV

Quadrant IV

270° to 360° (3π/2 to 2π)

Only cosine positive

Reference Angle Formulas

Degrees:

Q1: θ

Q2: 180° - θ

Q3: θ - 180°

Q4: 360° - θ

Radians:

Q1: θ

Q2: π - θ

Q3: θ - π

Q4: 2π - θ

Key Points

Reference angles are always acute (0° to 90°)

They help find trigonometric function values

ASTC rule: All Students Take Calculus

Co-terminal angles have the same reference angle

Understanding Reference Angles

What is a Reference Angle?

A reference angle is the acute angle (between 0° and 90°) formed between the terminal side of an angle in standard position and the x-axis. It's the smallest positive angle you can make with the x-axis.

Why are Reference Angles Important?

  • Simplify trigonometric calculations
  • Find exact values for trigonometric functions
  • Determine signs of trigonometric functions
  • Solve trigonometric equations efficiently

How to Use Reference Angles

Step 1: Normalize the angle to 0-360° (or 0-2π radians)

Step 2: Determine which quadrant the angle is in

Step 3: Apply the appropriate formula for that quadrant

Step 4: The result is always between 0° and 90°

ASTC Memory Aid: "All Students Take Calculus" helps remember which trigonometric functions are positive in each quadrant.