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

Practice problems

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: If sin^2(x) = 0.25, find cos^2(x).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Subtract from 1.
  4. Check the result by substituting or estimating: the response should match 0.75 and make sense in the original problem.

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. Warm-up: First identify exactly what the question is asking: tan(x) equals...
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Tangent is sine over cosine.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: sin(x)/cos(x)

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: If cos^2(x) = 0.81, find sin^2(x).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. 1 - 0.81 = 0.19.
  4. Check the result by substituting or estimating: the response should match 0.19 and make sense in the original problem.

Answer: 0.19

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

Choices: 1 · 0 · tan(x) · 2

Show solution
  1. Challenge: First identify exactly what the question is asking: sin^2(x) + cos^2(x) simplifies to...
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. This is the Pythagorean identity.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 1

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

Show solution
  1. Review: First identify exactly what the question is asking: If sin(x) = 4/5 and cos(x) = 3/5, what is sin^2(x) + cos^2(x)?
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Square each value: (4/5)^2 = 16/25 and (3/5)^2 = 9/25.
  4. Add 16/25 + 9/25 = 25/25.
  5. The sum is 1.
  6. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

Practice this interactively with instant feedback and an AI tutor.

Practice Trigonometric Identities Take the free placement check

More Algebra II lessons