Cosine of a Difference
Angle subtraction formula for cosine.
Formula
Variables
How it works
This identity tells you how to find the cosine of an angle formed by subtracting one angle from another without first calculating the subtraction directly. It comes from the way cosine and sine components combine when rotating directions in the plane, so the result is built from both the cosine and sine of each angle. The key point is that cosine of a difference is not just the difference of cosines; the sine terms are essential to preserve the correct geometry.
When to use it
Use this when you need to simplify or evaluate cos(A-B) in trigonometric proofs, exact-value calculations, or formula manipulations. It is also useful in physics and engineering whenever a phase shift or angle offset appears.
Worked example, step by step
Find cos(75 b0 - 15 b0). Step 1: Apply the identity: cos(A-B) = cos A cos B + sin A sin B Step 2: Substitute A = 75 b0 and B = 15 b0 cos(75 b0 - 15 b0) = cos 75 b0 cos 15 b0 + sin 75 b0 sin 15 b0 Step 3: Use the exact values by noticing 75 b0 - 15 b0 = 60 b0, so the expression should equal cos 60 b0 Step 4: cos 60 b0 = 1/2 Final answer: cos(75 b0 - 15 b0) = 1/2
Common mistakes to avoid
- Replacing cos(A-B) with cos A - cos B, which is not a valid identity
- Forgetting the plus sign between the two products in the formula
- Mixing up the order of the angles and using the subtraction formula for sine instead
Did You Know?
Ptolemy tabulated chords equivalent to these formulas around 150 AD to predict planetary motion.
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More in Trigonometry
View allSine Ratio (SOH)
BasicRatio of the side opposite an angle to the hypotenuse.
Cosine Ratio (CAH)
BasicRatio of the side adjacent to an angle to the hypotenuse.
Tangent Ratio (TOA)
BasicRatio of the opposite side to the adjacent side.
Cosecant
IntermediateReciprocal of sine.
Secant
IntermediateReciprocal of cosine.
Cotangent
IntermediateReciprocal of tangent.