Angle Addition Formulas- Trigonometry Essentials Explained
What Angle Addition Formulas Actually Are
Angle addition formulas let you break down the sine, cosine, and tangent of (A + B) into functions of A and B separately. That's it. That's the whole point.
Instead of memorizing values for weird angles, you combine angles you do know. For example, 75° = 45° + 30°. Now you can find sin(75°) without touching a calculator.
These formulas are the foundation for half-angle, double-angle, and just about everything else in trig. If you're shaky on these, you're going to struggle with the rest.
The Three Formulas You Need
Sine Addition Formula
sin(A + B) = sin A cos B + cos A sin B
Memorize it as: sin of the sum equals sin first times cos second plus cos first times sin second. The pattern is sin·cos + cos·sin.
Cosine Addition Formula
cos(A + B) = cos A cos B − sin A sin B
Pattern here: cos·cos minus sin·sin. Notice the minus sign—this is where most people mess up.
Tangent Addition Formula
tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
The denominator has a minus sign again. Don't forget it.
How to Actually Use These
Here's the process:
- Identify your target angle as a sum of angles you know
- Apply the correct formula
- Plug in values for sin, cos, or tan of the individual angles
- Simplify
Working Example: Find sin(75°)
Step 1: 75° = 45° + 30°
Step 2: sin(75°) = sin(45° + 30°)
Step 3: Apply the formula
sin(75°) = sin 45° cos 30° + cos 45° sin 30°
Step 4: Plug in known values
= (√2/2)(√3/2) + (√2/2)(1/2)
= √6/4 + √2/4
= (√6 + √2)/4
That's your answer. No calculator needed.
Working Example: Find cos(15°)
15° = 45° − 30°, so use cos(A − B) = cos A cos B + sin A sin B
cos(15°) = cos 45° cos 30° + sin 45° sin 30°
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√6 + √2)/4
Same result. Makes sense since sin(75°) = cos(15°).
Quick Reference Table
| Formula | Expression |
|---|---|
| Sine of sum | sin A cos B + cos A sin B |
| Sine of difference | sin A cos B − cos A sin B |
| Cosine of sum | cos A cos B − sin A sin B |
| Cosine of difference | cos A cos B + sin A sin B |
| Tangent of sum | (tan A + tan B) / (1 − tan A tan B) |
| Tangent of difference | (tan A − tan B) / (1 + tan A tan B) |
The difference formulas just flip the sign of the second term. Sine and tangent flip the middle operation, cosine flips the sign between terms.
Common Mistakes
- Forgetting the sign change in cosine formulas. cos(A + B) has a minus sign, not a plus.
- Using the wrong formula for sine vs. cosine. They look similar but the operations differ.
- Not simplifying at the end. Rationalize denominators if your answer has a fraction with √2 or √3 in the denominator.
- Assuming degrees vs. radians. Pick one and stick with it throughout the problem.
Where These Show Up Next
Angle addition formulas lead directly to:
- Double-angle formulas — set B = A to get sin(2A), cos(2A), tan(2A)
- Half-angle formulas — solve for sin(A/2) or cos(A/2)
- Proving trig identities — half of all identity problems use these
- Physics and engineering — vector addition, wave interference, rotations
Master these now or you'll be relearning them every time you hit a new trig topic. They're not going away.