Section 7.2

Sum & Difference Identities

Sum and difference identities let you compute exact values for angles like $75°$ or $\frac{\pi}{12}$ that aren't on the unit circle. They're also the foundation for the double-angle formulas in Section 7.3. Expect 1–2 'find the exact value' problems on Exam 3.

⚡ Quick Summary

What if you need the exact sine of 75°? It's not on the unit circle — but you can split it into two angles that are: $75° = 45° + 30°$ . The sum and difference formulas let you calculate exact trig values for these "in-between" angles by combining values you already know. The key thing to accept upfront: $\sin(A + B)$ is not $\sin(A) + \sin(B)$ . There's a specific formula you have to use.

Sine of a Sum

\sin(A + B) = \sin A \cos B + \cos A \sin B

Sine of a Difference

\sin(A - B) = \sin A \cos B - \cos A \sin B

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

The formulas follow a pattern. For sine, the functions alternate: $\sin(A + B) = \sin A \cos B + \cos A \sin B$ — think "sine-cosine-cosine-sine." For cosine, each function sticks with its own kind: $\cos(A + B) = \cos A \cos B - \sin A \sin B$ . The sneaky part: cosine uses the opposite sign — $\cos(A + B)$ has a minus, and $\cos(A - B)$ has a plus. It's the reverse of what you'd guess.

The most useful angle splits to memorize: $75° = 45° + 30°$ , $15° = 45° - 30°$ , $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$ , and $\frac{7\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}$ . If the problem gives you $\sin A$ and $\cos B$ with quadrant info instead of angles, just find the missing trig values using $\sin^2 + \cos^2 = 1$ before plugging into the formula.

This is one of the formula-heaviest sections — don't try to memorize everything at once. Focus on the sine and cosine sum formulas first, and remember: "sine-cosine-cosine-sine" for the sine formula, and "cosine is contrary" for the sign in the cosine formula.

Key Formulas

Sine of a Sum

\sin(A + B) = \sin A \cos B + \cos A \sin B

Sine of a Difference

\sin(A - B) = \sin A \cos B - \cos A \sin B

Cosine of a Sum

\cos(A + B) = \cos A \cos B - \sin A \sin B

Watch the sign — it's minus for the sum, which is the opposite of what you might guess.

Cosine of a Difference

\cos(A - B) = \cos A \cos B + \sin A \sin B

Tangent of a Sum/Difference

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

The sign in the denominator is opposite to the sign in the numerator.

Key Takeaways

✓ $\sin(A + B) \neq \sin A + \sin B$ — you must use the formula: $\sin A \cos B + \cos A \sin B$
✓Cosine formulas have the opposite sign: $\cos(A - B)$ has a plus, $\cos(A + B)$ has a minus
✓Key decompositions to memorize: $75° = 45° + 30°$ , $15° = 45° - 30°$ , $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$
✓If given $\sin A$ and $\cos B$ with quadrant info, find the missing values via $\sin^2 + \cos^2 = 1$ first

⚠️ Common Mistakes

✗Thinking $\sin(A + B) = \sin A + \sin B$ — this is wrong; you must use the full formula $\sin A\cos B + \cos A\sin B$
✗Getting the sign wrong in the cosine formula — $\cos(A + B)$ uses minus and $\cos(A - B)$ uses plus, which is the opposite of what you'd guess
✗Not knowing how to decompose non-standard angles — memorize these: $75° = 45° + 30°$ , $15° = 45° - 30°$ , $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$
✗Forgetting to find missing trig values before plugging in — if given $\sin A$ and a quadrant, find $\cos A$ first via $\sin^2 + \cos^2 = 1$

Next: Section 7.3 Builds on: 7.1

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7.1 Trig Identities

7.3 Double-Angle & Half-Angle Formulas