Double Angle Examples
Practice-friendly sine, cosine, and tangent double-angle examples in degrees and radians.
Quick Answer
- Short definition
- Double-angle examples substitute specific θ values into sin(2θ), cos(2θ), or tan(2θ) identities.
- Formula
- sin(2θ) = 2 sin θ cos θ
Introduction
Use the Double Angle Calculator alongside these examples to compare manual steps with live results. Worked examples turn abstract identities into numbers you can predict before you calculate.
This article collects sine, cosine, and tangent problems in both degrees and radians. Each example shows the identity used, the substitution step, and a direct check at 2θ when that check is quick.
Before diving in, skim the Double Angle Formula list so you know which cosine form fits each problem. If any step feels unfamiliar, the full calculation workflow in How to Calculate Double Angles breaks the process into smaller stages.
Main Content
What is it?
Examples turn identities into predictable numbers. Start with standard angles from the unit circle: 30°, 45°, 60°, π/6, π/4, and π/3. These angles give exact ratios with radicals, which makes manual verification clean.
Each example follows the same rhythm. Identify θ, pick the target function of 2θ, choose an identity, substitute, simplify, and confirm. Repeating that rhythm on familiar angles builds the muscle memory you need for exam problems with less friendly values.
Degree examples match how many high-school courses introduce the topic. Radian examples align with calculus and university trigonometry. Practice both so unit switches on tests do not slow you down.
Some examples rewrite products such as sin θ cos θ rather than evaluate sin(2θ) directly. That form appears often in integration exercises and is worth treating as its own pattern.
Formula
Keep all three core formulas visible while you work:
- sin(2θ) = 2 sin θ cos θ
- cos(2θ) = cos² θ − sin² θ = 2 cos² θ − 1 = 1 − 2 sin² θ
- tan(2θ) = 2 tan θ / (1 − tan² θ)
The formula guide explains when to pick each cosine version. In examples below, we call out the form used so you can see the decision in context.
Step-by-step guide
- Pick θ from a known ratio table. Use unit-circle values or given triangle ratios.
- Apply the identity for the target function. Write the identity before substituting numbers to avoid mixing sine and cosine forms.
- Simplify radicals and fractions. Leave answers in exact form unless decimals are requested.
- Enter the same θ in the calculator. Confirm all three outputs when learning a new angle.
- Review mistakes against the workflow. If results disagree, walk back through How to Calculate Double Angles to find whether the error is in the identity choice, the unit, or the arithmetic.
Example
Sine: degrees: θ = 30° → 2θ = 60°.
sin(60°) = 2 sin(30°) cos(30°) = 2(1/2)(√3/2) = √3/2.
Cosine: degrees: θ = 60° → 2θ = 120°.
cos(120°) = 2 cos²(60°) − 1 = 2(1/4) − 1 = −1/2.
Tangent: radians: θ = π/4 → 2θ = π/2.
tan(π/2) is undefined because cos(π/2) = 0. The formula denominator 1 − tan²(π/4) = 1 − 1 = 0 flags the same issue.
Product rewrite: sin(20°) cos(20°) = sin(40°)/2. This uses the sine identity divided by 2 and is a common intermediate step in calculus.
Radian sine: θ = π/6 → sin(π/3) = 2(1/2)(√3/2) = √3/2.
FAQ
Which angles are best for beginners?
Start with 30°, 45°, and 60°, then practice π/6, π/4, and π/3 in radians. These give exact values without a calculator.
Should I memorize the examples or the method?
Memorize the method. Examples change on every test; the identity selection and substitution steps stay the same.
Conclusion
Repeat each example until you can predict the sign and size before calculating. Speed comes from recognizing which identity fits, not from memorizing final decimals.
When you are ready to simplify expressions rather than evaluate numeric angles, move to Double Angle Identities for transformation-focused practice.
Practice now with the same θ values from this page.