Sum and Difference Identities

Functions
principle

Also known as: angle addition formulas, sum-difference formulas

Grade 9-12

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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. 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.

Definition

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.

💡 Intuition

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.

🎯 Core Idea

These formulas decompose trig functions of combined angles into products of trig functions of individual angles. They are the foundation for double-angle, half-angle, and product-to-sum formulas.

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}

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}

Notation

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

🌟 Why It 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.

💭 Hint When Stuck

Verify by plugging in known angles: check that cos(60) = cos(30+30) gives the right answer using the formula.

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}

🚧 Common Stuck Point

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.

⚠️ 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.

Frequently Asked Questions

What is Sum and Difference Identities in Math?

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.

Why is Sum and Difference Identities important?

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 usually 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 Sum and Difference Identities?

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

How Sum and Difference Identities Connects to Other Ideas

To understand sum and difference identities, you should first be comfortable with trig identities pythagorean and trigonometric functions. Once you have a solid grasp of sum and difference identities, you can move on to trig identities double angle.

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