Double Angle Identities- Khan Academy Trigonometry Tutorial

What Are Double Angle Identities?

Double angle identities are formulas that express trigonometric functions of 2θ in terms of functions of θ. They're not new math—they're shortcuts. Once you know the basic trig ratios, these identities let you simplify expressions and solve problems without starting from scratch every time.

You encounter them in calculus, physics, engineering, and anywhere angles repeat or combine. Khan Academy covers these in their trigonometry unit, but the explanations there can feel thin if you want actual understanding rather than memorization.

The Core Formulas

Three identities matter. Memorize these or know how to derive them fast.

Sine Double Angle

sin(2θ) = 2 sinθ cosθ

This is the simplest one. It comes straight from the sum identity for sine: sin(θ + θ) = sinθ cosθ + cosθ sinθ. That's it. Two identical terms, combined.

Cosine Double Angle

The cosine identity has three forms. This trips people up because you need to know all of them:

• cos(2θ) = cos²θ − sin²θ
• cos(2θ) = 2cos²θ − 1
• cos(2θ) = 1 − 2sin²θ

These are all equivalent. The second and third forms are useful when you need to convert between sine and cosine squared terms. Pick whichever makes your algebra cleaner.

Tangent Double Angle

tan(2θ) = 2tanθ / (1 − tan²θ)

This works when tanθ is defined. If tan²θ = 1, the denominator becomes zero and the identity breaks. That's a domain restriction you need to remember.

When to Use Each Form

Most students memorize everything and then freeze up when problems don't look exactly like the formulas. Here's when each form actually helps:

How to Derive the Identities (And Why You Should)

Deriving these takes 30 seconds and saves you from memorizing them wrong. Here's the sine derivation:

Start with the sum identity:

sin(A + B) = sinA cosB + cosA sinB

Set A = B = θ:

sin(θ + θ) = sinθ cosθ + cosθ sinθ

sin(2θ) = 2sinθ cosθ ✓

For cosine, use the same approach:

cos(A + B) = cosA cosB − sinA sinB

Set A = B = θ:

cos(2θ) = cos²θ − sin²θ ✓

From cos²θ − sin²θ, you get the other forms using the Pythagorean identity (sin²θ + cos²θ = 1). Solve for cos²θ or sin²θ and substitute.

Practical Examples

Example 1: Find sin(2θ) Given sinθ

Problem: If sinθ = 3/5 and θ is in Quadrant I, find sin(2θ).

You need cosθ first. Draw a right triangle: opposite = 3, hypotenuse = 5. Adjacent = √(5² − 3²) = √16 = 4.

cosθ = 4/5

sin(2θ) = 2sinθ cosθ = 2(3/5)(4/5) = 24/25

Done. No memorization of special angles needed.

Example 2: Simplify cos(2θ) Using the Right Form

Problem: Simplify 1 − 2sin²(3x)

Recognize the pattern: 1 − 2sin²θ looks like the cosine double angle form.

Using cos(2θ) = 1 − 2sin²θ:

1 − 2sin²(3x) = cos(2 · 3x) = cos(6x)

That's it. Two seconds if you know the form.

Example 3: Verify an Identity

Problem: Verify that (1 + tanθ) / (1 − tanθ) = tan(π/4 + θ)

This uses the tangent sum formula, which you can derive or look up. The key insight: tan(π/4 + θ) = (tanπ/4 + tanθ) / (1 − tanπ/4 · tanθ) = (1 + tanθ) / (1 − tanθ).

The left side matches. Identity verified. Double angle identities show up constantly in verification problems because you can rewrite expressions in multiple equivalent forms.

Getting Started on Khan Academy

Khan Academy's trigonometry section covers double angle identities, but the progression isn't always clear. Here's how to use it effectively:

• Start with the Sum and Difference Identities unit. Double angle formulas are just special cases of these.
• Watch the derivation videos, not just the formula videos. Understanding where formulas come from matters more than memorizing them.
• Complete the Practice: Double Angle Identities section. The problems repeat the same patterns until you recognize them.
• Don't skip the Proofs exercises. They're annoying but they force you to internalize the logic.

Common Mistakes

• Using the wrong cosine form. Students pick cos(2θ) = cos²θ − sin²θ when they should use 1 − 2sin²θ for the problem at hand. Check what you're trying to eliminate.
• Forgetting the domain restriction on tangent. When tan²θ = 1, the denominator in tan(2θ) = 2tanθ / (1 − tan²θ) equals zero. The identity doesn't apply at θ = π/4 + nπ/4.
• Mixing up half-angle and double angle. These are inverse operations. Don't confuse sin(2θ) with sin(θ/2). They use completely different formulas.
• Memorizing without understanding. You will forget. Deriving the identities takes 30 seconds and you can reconstruct them on the spot.

Quick Reference Table

Bottom Line

Double angle identities aren't difficult. They're just sum identities with identical angles. Learn the derivations, know which cosine form to use, and practice recognizing patterns. Khan Academy has enough practice problems to make these automatic—just work through them until you stop having to think.