Addition and Subtraction Formulas
The unit circle hands you exact values at 30, 45, 60, and a few other angles. So what is the exact value of cos 15 degrees? Fifteen is not on the circle, but it is the difference 45 minus 30, and the addition and subtraction formulas turn that difference into something you can evaluate by hand.
By the end of this page you can:
- State the addition and subtraction formulas for sine, cosine, and tangent.
- Find exact values such as cos 15 degrees by writing the angle as a sum or difference of special angles.
- Use the formulas as a step inside an identity proof.
The floor. Memorize the cosine formula cos(a + b) = cos a cos b minus sin a sin b, watch the sign, and never split sin(a + b) into sin a + sin b.
Quick Reference
| Field | Value |
|---|---|
| Textbook | Stewart, Redlin, Watson, Precalculus 7th ed. |
| Chapter | 7 |
| Section | 7.2 Addition and Subtraction Formulas |
| Open alternate | OpenStax Precalculus 2e, 9.2 Sum and Difference Identities |
| Difficulty | Core |
| Time | About 30 minutes |
Before You Start
The Idea
A trig function of a sum is not the sum of the trig functions. That single fact causes most of the errors here. The value of sin(a + b) depends on all four pieces sin a, cos a, sin b, and cos b, woven together in a fixed pattern. The same is true for cosine, with one extra trap: the cosine formula flips the sign, so a plus inside becomes a minus in the expansion.
Once you trust the pattern, any angle you can write as a sum or difference of special angles becomes exactly computable. The tangent formula follows from dividing the sine formula by the cosine formula.
Key Formulas
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
\[ \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \]
\[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]
Source: Stewart 7e, Section 7.2 Addition and Subtraction Formulas.
Notice the sign rule: for sine, the sign inside matches the sign in the middle of the expansion. For cosine, the sign inside is the opposite of the sign in the middle.
Worked Example
Find the exact value of cos 15 degrees.
Step 1. Write 15 degrees as a difference of special angles: 15 = 45 minus 30.
Step 2. Apply the cosine subtraction formula with a = 45 and b = 30: \[ \cos(45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ \]
Step 3. Substitute the unit-circle values cos 45 = √2 over 2, cos 30 = √3 over 2, sin 45 = √2 over 2, and sin 30 = 1 over 2.
Show the answer
\[ \cos 15^\circ = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2}\cdot\frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \]
As a decimal this is about 0.9659, which matches a calculator value of cos 15 degrees. Writing 15 as 60 minus 45 gives the same answer.
Connections
This builds on proving and simplifying trigonometric identities and on reading exact sine and cosine values from the unit circle. The cosine subtraction formula is the engine behind the next step, the double-angle formulas, where you set both angles equal. The same machinery later produces the half-angle and product-to-sum formulas. Double-angle and half-angle formulas are separate topics and are not derived on this page.
Resources
- Primary: Stewart, Redlin, Watson, Precalculus 7th ed., Section 7.2 Addition and Subtraction Formulas.
- Alternate: OpenStax Precalculus 2e, Section 9.2 Sum and Difference Identities.
Check your understanding
Which expression equals cos(a + b)?
Last updated: 2026-06-24