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sin²x Is a Wave

Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 35 of 48.

${\cos\left(x\right)}^{2} = \frac{1}{2}+\frac{\cos\left(2x\right)}{2}$ . Read it as a graph: a cosine wave of height 1/2, twice the usual speed, lifted so that its midline is 1/2. It runs between 0 and 1, never below 0, which is right for a square.

$The graph of cos squared x, a wave between 0 and 1$

y = cos²x: midline 1/2, height 1/2, one wave every π.

The square of a wave is a wave. Its midline is the average of the square, 1/2, and its period is half the original, because squaring makes the negative half of each wave look like the positive half.

Challenge

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

Try it yourself

y = \sin^{2} x

Draw the wave

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 Double The Graph of 2 sin x cos x Recap: 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 Reductionsin²x Is a WaveHalf-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