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Reading Signs on the Circle

Part of the Trig Equations and Inverses section of Coddy's Trigonometry journey. Lesson 2 of 52.

Ask a calculator for the angle with sine −1/2. It says −30°. But −30° is not between 0° and 360°. So which angles are?

−30° is not the answer, but it helps. Its size, 30°, is the reference angle: the angle to the nearest part of the x-axis. The minus sign means below the x-axis, in quadrants III and IV. So the angles are $180^{\circ}+30^{\circ}$ , which is 210°, and $360^{\circ}-30^{\circ}$ , which is 330°. The calculator gives the reference angle. The sign tells you the quadrants.

Challenge

A calculator says $\arcsin\left(\frac{-\sqrt{3}}{2}\right)$ is −60°. Mark every angle θ from 0° up to 360° with $\sin\left(\theta\right) = \frac{-\sqrt{3}}{2}$ .

Then press Check.

Start the lesson to read it in the interactive player and track your progress.

All lessons in Trig Equations and Inverses

1Your First Trig Equations

One Equation, Two AnglesReading Signs on the CircleRecap: First Equations

4Compositions

Draw the Triangle The Outer Function A Negative Inside When Undoing Does Not Undo Recap: Compositions

7General Solutions

Every Solution Tangent's Single Family Families in Radians Families, Rounded Recap: Every Solution

10Identities First

Write sin 2x Out Pick the Right cos 2x Swap cos²x for sin²x sec, csc and cot Recap: Identities First

2Inverses and Their Ranges

Why arcsin Gives One Angle The Range of arccos The Range of arctan Recap: Inverse Ranges

5Equations on a Domain

A Negative Cosine Tangent Equations Equations in Radians Two Turns, Four Answers Recap: On a Domain

8Multiple Angles

Two Turns for 2x Solving sin 2x Half Angles Every Solution of sin 3x Recap: Multiple Angles

11Final Challenges

Final: The Bicycle Pedal Final: The Big Wheel Final: Every Solution Final: An Identity and More Final: An Inverse Inside

3Evaluating Inverse Trig

Rounded Angles in Radians An Angle From a Slope Recap: Inverse Values

6Off the Special Angles

Both Answers, Rounded A Negative Sine, Rounded Cosine, Rounded Tangent, Rounded Rounded, in Radians Recap: Off the Table

9Factoring Trig Equations

A Common Factor Never Divide by sin x A Quadratic in Sine A Case With No Angle Square Roots Give Two Signs Recap: Factoring