The Graph of 2 sin x cos x

Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 32 of 48.

The double-angle formulas turn some products and squares of waves into a single wave. ${\cos\left(x\right)}^{2}-{\sin\left(x\right)}^{2}$ is $\cos\left(2x\right)$ : a cosine squeezed to twice the speed, one full wave every π.

$The graph of cos 2x, period π$

y = cos²x - sin²x is y = cos 2x.

Doubling the angle halves the period: 2π becomes π. The height does not change, because the 2 sits inside the angle, not in front of the function.

Challenge

Draw the graph of $y = 2\sin\left(x\right)\cos\left(x\right)$ . Then press Check.

Try it yourself

y = 2\sin x \cos x

Draw the wave

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Trig Identities

1Identities on the Circle

sin²x + cos²x = 1 Squares Erase the Signs Recap: the Circle Identity

4Even, Odd and Cofunctions

Negative Angles Even and Odd in a Proof Even and Odd Graphs Cofunctions Recap: Signs of Angles

7Double-Angle Formulas

sin 2x = 2 sin x cos x Three Forms of cos 2x Exact Values, Doubled Square, Then DoubleThe Graph of 2 sin x cos xRecap: Double Angles

10Final Challenges

Final: Two Angles Final: Tangents to sin 2x Final: An Exact Product Final: A Wave in Disguise Final: Two Squares Final: Choose the Form

2Reciprocal and Quotient

Identity or Equation?Secant, Cosecant, Cotangent Recap: Reciprocals

5Proving Identities

Work on One Side Rewrite in sin and cos Factor First Make One Fraction Proofs That Go Wrong Recap: Prove It

8Half Angles, Power Reduction

Power Reduction sin²x Is a Wave Half-Angle Values tan of a Half Angle Recap: Half Angles

3Pythagorean Identities

1 + tan²x = sec²x A Difference of Squares Squaring a Sum Recap: Three Forms

6Sum and Difference

sin(A + B) and sin(A - B)cos(A + B) and cos(A - B)Turning by 90° and 180°Read the Formula Backwards tan(A + B) and tan(A - B)Recap: Sum Formulas

9Product to Sum

Products to Sums Sums to Products Sums Over Sums Recap: Sum to Product