Double-Angle and Half-Angle Identities
A free Trigonometry lesson from the “Trig Identities” unit, with a worked example and practice problems including step-by-step solutions.
Trig identities are equivalence tools. Students rewrite expressions with reciprocal, quotient, Pythagorean, sum, difference, double-angle, and half-angle identities while preserving valid algebraic steps.
What you'll learn
- Apply reciprocal, quotient, and Pythagorean identities
- Simplify and verify identities without invalid algebra
- Use sum, difference, double-angle, and half-angle formulas
Worked example
Problem. Double-Angle and Half-Angle Identities: If sin(theta)=3/5 and theta is in Quadrant I, find cos(theta).
- Use sin^2 + cos^2 = 1.
- cos^2 = 16/25.
- Cosine is positive, so cos = 4/5.
Answer: 4/5
Practice problems
1. Double-Angle and Half-Angle Identities: Rewrite tan(x) using sine and cosine.
Show solution
- Warm-up: First identify exactly what the question is asking: Double-Angle and Half-Angle Identities: Rewrite tan(x) using sine and cosine.
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Use the quotient identity.
- Tangent equals sine divided by cosine.
- tan(x) = sin(x)/cos(x).
- Check the result by substituting or estimating: the response should match sin(x)/cos(x) and make sense in the original problem.
Answer: sin(x)/cos(x)
2. Double-Angle and Half-Angle Identities: Simplify 1 - sin^2(x).
Show solution
- Warm-up: First identify exactly what the question is asking: Double-Angle and Half-Angle Identities: Simplify 1 - sin^2(x).
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Use sin^2(x) + cos^2(x) = 1.
- Subtract sin^2(x).
- The result is cos^2(x).
- Check the result by substituting or estimating: the response should match cos^2(x) and make sense in the original problem.
Answer: cos^2(x)
3. Double-Angle and Half-Angle Identities: Simplify tan(x)cos(x).
Show solution
- Warm-up: First identify exactly what the question is asking: Double-Angle and Half-Angle Identities: Simplify tan(x)cos(x).
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Replace tangent with sin(x)/cos(x).
- Cancel the cosine factor.
- The result is sin(x).
- Check the result by substituting or estimating: the response should match sin(x) and make sense in the original problem.
Answer: sin(x)
4. Double-Angle and Half-Angle Identities: If sin(theta)=3/5 and theta is in Quadrant I, find cos(theta).
Show solution
- Core Practice: First identify exactly what the question is asking: Double-Angle and Half-Angle Identities: If sin(theta)=3/5 and theta is in Quadrant I, find cos(theta).
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Use sin^2 + cos^2 = 1.
- cos^2 = 16/25.
- Cosine is positive, so cos = 4/5.
- Check the result by substituting or estimating: the response should match 4/5 and make sense in the original problem.
Answer: 4/5
5. Double-Angle and Half-Angle Identities: Which identity is sin(a + b)?
Choices: sin(a)cos(b) + cos(a)sin(b) · cos(a)cos(b) - sin(a)sin(b) · tan(a) + tan(b) · sin(a)sin(b)
Show solution
- Core Practice: First identify exactly what the question is asking: Double-Angle and Half-Angle Identities: Which identity is sin(a + b)?
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Use the sine sum identity.
- It combines sine-cosine products.
- The first choice is correct.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: sin(a)cos(b) + cos(a)sin(b)
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