Half Angles

Part of the Trig Equations and Inverses section of Coddy's Trigonometry journey. Lesson 34 of 52.

A half angle runs the other way. In $2\sin\left(\frac{x}{2}\right) = 1$ , while x runs over [0°, 360°), x/2 runs over [0°, 180°): only half a turn. sin(x/2) = 1/2 at x/2 = 30° or 150°, both in that half turn, so x = 60° or 300°.

A half turn can also hold fewer answers than a full one. cos(x/2) = −1/2 needs x/2 = 120° or 240°, and only 120° is below 180°: one answer, x = 240°.

Challenge

Solve $2\cos\left(\frac{x}{2}\right) = \sqrt{2}$ for every x in [0°, 360°), then press Check.

Try it yourself

2\cos\left(\frac{x}{2}\right) = \sqrt{2}

Get $x$ alone $0^{\circ} \le x < 360^{\circ}$

All lessons in Trig Equations and Inverses

1Your First Trig Equations

One Equation, Two Angles Reading Signs on the Circle Recap: 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 2xHalf AnglesEvery 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