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Two Turns for 2x

Part of the Trig Equations and Inverses section of Coddy's Trigonometry journey. Lesson 36 of 56.

In cos(2x) = 0 the angle inside is 2x, not x. While x runs over [0°, 360°), 2x runs over [0°, 720°): two full turns. So solve for u = 2x over two turns first.

cos u = 0 at 90° and 270° in the first turn, and 450° and 630° in the second. Now halve each: x = 45°, 135°, 225° or 315°. Halving only the first turn's angles gives two answers and misses half of them.

$The graph of cos 2x over one turn, crossing zero four times$

cos 2x makes two full waves while x makes one turn.

Challenge

Solve sin 2x = 1/2 for x in [0°, 360°), first step. Let u = 2x, so u runs over [0°, 720°). Tap every u in [0°, 720°) with sin(u) = 1/2, using the Turn switch, and 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 Trig Equations and Inverses

1Your First Trig Equations

One Equation, Two Angles Reading Signs on the Circle The Reference Angle 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 Negative Inputs to arcsin 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 2xSolving 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 Degrees Rounded Angles in Radians arctan, Rounded 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