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Reference in III and IV

Part of the Angles and the Unit Circle section of Coddy's Trigonometry journey. Lesson 31 of 57.

In quadrant III the side has gone past the negative x-axis, so the reference angle is how far past 180° it went: for 225° that is 225° - 180°, which is 45°.

$The terminal side of 330° in quadrant IV, 30° short of a full turn$

In quadrant IV the side is closing in on the positive x-axis, so the reference angle is what is left to a full turn: for 330° that is 360° - 330°, which is 30°. A negative angle works the same once you see where it ends: −30° ends in quadrant IV, 30° below the axis.

So for an angle from 0° to 360° the three rules are one idea: measure the gap to the nearer end of the x-axis, 180° in quadrants II and III, 360° in quadrant IV. In quadrant I the angle is its own reference angle.

Challenge

Choose the reference angle of 240°, 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 IVSigns 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