Product: sin A cos B
Product-to-sum identity.
Formula
Variables
How it works
This identity rewrites a product of a sine and a cosine as the average of two sine waves: one with the sum of the angles and one with their difference. It comes from the sine addition and subtraction formulas, since adding those two equations makes the cosine terms combine into a product. The result is especially useful because it converts multiplication into addition, which often simplifies algebra, integration, and signal analysis.
When to use it
Use this when you need to simplify trigonometric expressions containing a sine times a cosine, especially inside integrals or Fourier-style calculations. It is also helpful when you want to transform an oscillation product into separate frequencies or to rewrite expressions in a form that is easier to compare or integrate.
Worked example, step by step
Evaluate \(\sin 75^\circ \cos 15^\circ\). Use the identity: \(\sin A\cos B = \tfrac{1}{2}[\sin(A+B)+\sin(A-B)]\) Substitute \(A=75^\circ\), \(B=15^\circ\): \(\sin 75^\circ\cos 15^\circ = \tfrac{1}{2}[\sin(90^\circ)+\sin(60^\circ)]\) Compute the sine values: \(\sin(90^\circ)=1\), \(\sin(60^\circ)=\tfrac{\sqrt{3}}{2}\) So: \(\sin 75^\circ\cos 15^\circ = \tfrac{1}{2}\left(1+\tfrac{\sqrt{3}}{2}\right)\) \(= \tfrac{1}{2}+\tfrac{\sqrt{3}}{4}\) Final answer: \(\tfrac{2+\sqrt{3}}{4}\)
Common mistakes to avoid
- Forgetting the factor of 1/2 in front of the bracket
- Using \(A-B\) with the wrong sign or swapping the two angles incorrectly
- Applying the identity to \(\sin A\sin B\) or \(\cos A\cos B\), which require different product-to-sum formulas
Did You Know?
Product-to-sum formulas are the mathematical heart of how radios mix and tune signals.
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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.