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Reference Angles

Part of the Angles and the Unit Circle section of Coddy's Trigonometry journey. Lesson 30 of 57.

The reference angle of an angle is the acute angle between its terminal side and the x-axis. For an angle inside a quadrant it is between 0° and 90°, and it is always measured to the x-axis, never to the y-axis.

$The terminal side of 150° in quadrant II and the side of 30° in quadrant I$

150° stops 30° short of the negative x-axis: its reference angle is 30°.

Take 150°. It ends in quadrant II, 30° short of the negative x-axis at 180°, so its reference angle is 180° - 150°, which is 30°. The triangle under its point is the 30° triangle mirrored, which is why the reference angle matters: it tells you which special triangle sits under the point. In quadrant II the reference angle is 180° − θ.

Challenge

Choose the reference angle of 135° from the buttons under the circle, then press Check.

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Angles and the Unit Circle

1A Point on a Circle

Height on the Circle Across: the Cosine Same Height, Two Angles Recap: Height and Across

4Radians

What a Radian Is Degrees to Radians Radians to Degrees Radians Without π Turns in Radians Recap: Radians

7Reference Angles and Signs

Reference AnglesReference in III and IV Signs by Quadrant Sign and Size Signs at Any Angle Recap: Reference Angles

10Six Functions, Any Circle

Any Circle, Any Radius From a Point to the Angle Secant, Cosecant, Cotangent Reciprocals in Radians Recap: Six Functions

2Standard Position

Angles in Standard Position Negative Angles Quadrants and Axes Recap: Placing Angles

5Arc Length and Sector Area

Arc Length Arcs From Degrees Finding the Angle Sector Area Recap: A Slice of Pizza

8All Four Quadrants

Points in Quadrant II Quadrants III and IV Every Angle for a Value Points in Radians Exact Values in Radians Recap: Four Quadrants

11Final Challenges

The Minute Hand The Ferris Wheel Seat Four Angles, Two Turns The Garden Sprinkler A Negative Radian Point Which Angle?

3Coterminal Angles

Coterminal Angles Back Into One Turn Values Repeat Every Turn Every Turn at Once Recap: Coterminal Angles

6The Unit Circle

Why (cos θ, sin θ)The 30° and 60° Points The 45° Point Points Off the Table Recap: Quadrant I Points

9Tangent on the Circle

Tangent Is y over x Where Tangent Is Undefined Tangent in Radians Every Angle for a Tangent Recap: Tangent