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Values Repeat Every Turn

Part of the Angles and the Unit Circle section of Coddy's Trigonometry journey. Lesson 9 of 55.

$\sin\left(750^{\circ}\right)$ is exactly 1/2, the same as $\sin\left(30^{\circ}\right)$ . 750° is two full turns and 30° more. Full turns bring the point back to the same place.

Coterminal angles end at the same point, so they have the same sine and cosine: $\sin\left(\theta+360^{\circ}\right) = \sin\left(\theta\right)$ . It works the same way for cosine. Take off whole turns first. Then find the value.

Challenge

Work out $\sin\left(810^{\circ}\right)$ exactly, without a calculator.

Then press Check.

Try it yourself

\sin 810^{\circ}

Simplify

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 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 AnglesValues Repeat Every TurnEvery 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