CMClearMathAcademy

Trigonometric Identities

A free Algebra II lesson from the “Trigonometry and Modeling” unit, with a worked example and practice problems including step-by-step solutions.

Trig identities are equations true for all allowed angle values. The Pythagorean identity sin^2(x) + cos^2(x) = 1 is one of the most useful.

What you'll learn

Why it matters: Audio synthesis, surveying calculations, and physics derivations rewrite trig expressions using identities. The Pythagorean identity and the angle-sum identities are the workhorses — most problems become straightforward once the right one is applied.

Worked example

Problem. If sin^2(x) = 0.36, find cos^2(x).

  1. Use sin^2(x) + cos^2(x) = 1.
  2. 0.36 + cos^2(x) = 1.
  3. cos^2(x) = 0.64.

Answer: 0.64

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. If sin^2(x) = 0.25, find cos^2(x).

Show solution
  1. Subtract from 1.

Answer: 0.75

2. tan(x) equals...

Choices: sin(x)/cos(x) · cos(x)/sin(x) · sin(x) + cos(x) · 1 - cos(x)

Show solution
  1. Tangent is sine over cosine.

Answer: sin(x)/cos(x)

3. If cos^2(x) = 0.81, find sin^2(x).

Show solution
  1. 1 - 0.81 = 0.19.

Answer: 0.19

4. sin^2(x) + cos^2(x) simplifies to...

Choices: 1 · 0 · tan(x) · 2

Show solution
  1. This is the Pythagorean identity.

Answer: 1

5. If sin(x) = 4/5 and cos(x) = 3/5, what is sin^2(x) + cos^2(x)?

Show solution
  1. Square each value: (4/5)^2 = 16/25 and (3/5)^2 = 9/25.
  2. Add 16/25 + 9/25 = 25/25.
  3. The sum is 1.

Answer: 1

Practice this interactively with instant feedback and an AI tutor.

Practice Trigonometric Identities Take the free placement check

More Algebra II lessons