Cosine Angle Sum
Angle addition formula for cosine.
Formula
Variables
Example
cos 75° = cos(45°+30°) = cos45cos30 − sin45sin30 ≈ 0.259.
How it works
This identity tells you how to find the cosine of a sum or difference of two angles without needing to measure the combined angle directly. It comes from the geometry of the unit circle (or equivalently, from rotating a point by two angles in sequence), which reveals that the x-coordinate after two rotations depends on both cosine and sine terms. The "minus becomes plus" pattern is important: for cos(A + B), the sine-product term is subtracted, while for cos(A − B), it is added.
When to use it
Use this when an angle is built from simpler angles, especially common angles like 15°, 75°, 105°, or when simplifying trigonometric expressions in geometry and calculus. It is also useful for exact-value calculations and proofs involving angle decomposition.
Worked example, step by step
Find cos(15°). cos(15°) = cos(45° − 30°) Using cos(A − B) = cos A cos B + sin A sin B cos(15°) = cos45°cos30° + sin45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4 ≈ 0.966
Common mistakes to avoid
- Using the wrong sign: cos(A + B) needs a minus between the sine terms, while cos(A − B) needs a plus
- Forgetting to apply the formula to both terms, not just one of them
- Mixing up sine and cosine values for special angles such as 30° and 45°
Did You Know?
These sum formulas let you compute the sine or cosine of an angle by breaking it into known pieces.
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