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Turns in Radians

Part of the Angles and the Unit Circle section of Coddy's Trigonometry journey. Lesson 16 of 55.

In radians a full turn is $2\pi$ , so coterminal angles differ by multiples of $2\pi$ instead of 360°. $\frac{\pi}{4}$ and $\frac{9\pi}{4}$ end on the same side, because $\frac{9\pi}{4}-2\pi = \frac{\pi}{4}$ .

That makes repeating values easy to list. The height is −1 only at the bottom of the circle, $\frac{3\pi}{2}$ . In the second turn the bottom is reached again at $\frac{3\pi}{2}+2\pi$ , which is $\frac{7\pi}{2}$ . Each turn adds one more member to the family.

The sizes of the special angles in radians are worth knowing by sight: 30° is π/6, 45° is π/4, 60° is π/3, 90° is π/2.

Challenge

The circle is labelled in radians. Tap every angle from 0 up to $4\pi$ (two turns) where the height is one half, $\sin\left(\theta\right) = \frac{1}{2}$ . The Turn buttons above the circle switch between turns. Then press Check.

Start the lesson to read it in the interactive player 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 πTurns in RadiansRecap: Radians

7Reference Angles and Signs

Reference Angles Reference 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 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 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