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Exact Values, Doubled

Part of the Trig Identities section of Coddy's Trigonometry journey. Lesson 30 of 48.

Read backwards, the double-angle formulas turn an angle you do not know into its double, which you may know. Take $2\sin\left(15^{\circ}\right)\cos\left(15^{\circ}\right)$ . Its shape is 2 sin A cos A, so it is $\sin\left(2\cdot 15^{\circ}\right)$ , which is $\sin\left(30^{\circ}\right)$ , and that is 1/2.

The same works with any form of cos 2x. A line that reads cos²A - sin²A, or 2cos²A - 1, or 1 - 2sin²A, is cos 2A; double the angle and read the value off the special angles.

Challenge

Find the exact value of ${\cos\left(22.5^{\circ}\right)}^{2}-{\sin\left(22.5^{\circ}\right)}^{2}$ .

On this board the formula chip writes the values unworked; type each next line yourself in the box, and a line that does not equal the one above is refused.

Fold it into one cosine with the formula chip, type the cosine of the doubled angle, then its exact value. Then press Check.

Try it yourself

\cos^{2}\left(22.5^{\circ}\right)-\sin^{2}\left(22.5^{\circ}\right)

Rewrite as $\frac{\sqrt{2}}{2}$

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 2xExact Values, DoubledSquare, 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 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