Sum and Difference Identities Formula

The Formula

\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B
\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B
\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}

When to use: What happens when you combine two rotations? If you rotate by angle A and then by angle B, the result involves both angles interacting. The sum and difference formulas tell you exactly how the trig values of two separate angles combine. They're like a multiplication rule for rotations—the result isn't simply adding the trig values, but mixing sines and cosines together.

Quick Example

\cos(75°) = \cos(45° + 30°) = \cos 45°\cos 30° - \sin 45°\sin 30° = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}

Notation

Note the sign pattern: in the cosine formula, \pm becomes \mp (signs are opposite). In sine and tangent, signs match.

What This Formula Means

Formulas that express \sin(A \pm B), \cos(A \pm B), and \tan(A \pm B) in terms of \sin A, \cos A, \sin B, and \cos B.

What happens when you combine two rotations? If you rotate by angle A and then by angle B, the result involves both angles interacting. The sum and difference formulas tell you exactly how the trig values of two separate angles combine. They're like a multiplication rule for rotations—the result isn't simply adding the trig values, but mixing sines and cosines together.

Formal View

\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B; \cos(A \pm B) = \cos A\cos B \mp \sin A\sin B; \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}

Worked Examples

Example 1

easy
Find the exact value of \cos(75°) using the sum identity.

Solution

  1. 1
    Write 75° = 45° + 30°.
  2. 2
    Apply the cosine sum formula: \cos(A+B) = \cos A \cos B - \sin A \sin B.
  3. 3
    Substitute: \cos(75°) = \cos(45°)\cos(30°) - \sin(45°)\sin(30°).
  4. 4
    = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}.

Answer

\cos(75°) = \frac{\sqrt{6} - \sqrt{2}}{4}
The sum and difference identities let us find exact values for angles that are sums or differences of standard angles (30°, 45°, 60°). Here we decomposed 75° as 45° + 30° and applied the cosine addition formula.

Example 2

medium
Simplify \sin(x + y) + \sin(x - y).

Common Mistakes

  • Getting the cosine sign wrong: \cos(A + B) = \cos A\cos B \mathbf{-} \sin A\sin B (minus for addition, plus for subtraction—opposite of what feels natural).
  • Thinking \sin(A + B) = \sin A + \sin B—trig functions do NOT distribute over addition.
  • Forgetting to use the formulas when computing exact values: \sin(75°) requires the sum formula, not a calculator.

Why This Formula Matters

They enable computing exact trig values for non-standard angles (like 75° or \frac{\pi}{12}), proving other identities, and solving equations in physics and engineering involving combined oscillations.

Frequently Asked Questions

What is the Sum and Difference Identities formula?

Formulas that express \sin(A \pm B), \cos(A \pm B), and \tan(A \pm B) in terms of \sin A, \cos A, \sin B, and \cos B.

How do you use the Sum and Difference Identities formula?

What happens when you combine two rotations? If you rotate by angle A and then by angle B, the result involves both angles interacting. The sum and difference formulas tell you exactly how the trig values of two separate angles combine. They're like a multiplication rule for rotations—the result isn't simply adding the trig values, but mixing sines and cosines together.

What do the symbols mean in the Sum and Difference Identities formula?

Note the sign pattern: in the cosine formula, \pm becomes \mp (signs are opposite). In sine and tangent, signs match.

Why is the Sum and Difference Identities formula important in Math?

They enable computing exact trig values for non-standard angles (like 75° or \frac{\pi}{12}), proving other identities, and solving equations in physics and engineering involving combined oscillations.

What do students get wrong about Sum and Difference Identities?

The sign pattern in the cosine formula is opposite to what you might expect: \cos(A + B) has a MINUS sign between the terms, while \cos(A - B) has a PLUS sign.

What should I learn before the Sum and Difference Identities formula?

Before studying the Sum and Difference Identities formula, you should understand: trig identities pythagorean, trigonometric functions.

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