Product: sin A sin B
Product-to-sum identity.
Formula
Variables
How it works
This identity rewrites a product of two sine functions as a difference of two cosine functions. It comes from the cosine subtraction formula, because \(\cos(A-B)=\cos A\cos B+\sin A\sin B\) and \(\cos(A+B)=\cos A\cos B-\sin A\sin B\); subtracting these cancels the \(\cos A\cos B\) terms and isolates \(2\sin A\sin B\). So the product becomes a sum/difference form that is often easier to simplify, integrate, or analyze.
When to use it
Use this when you need to simplify a product of trig functions, especially in integration, signal processing, or proving identities. It is also useful when converting oscillating products into sums of waves with different frequencies.
Worked example, step by step
Evaluate \(\sin 75^\circ\sin 15^\circ\) using the product-to-sum identity. \(\sin A\sin B=\tfrac12[\cos(A-B)-\cos(A+B)]\) Let \(A=75^\circ\), \(B=15^\circ\). \(\sin 75^\circ\sin 15^\circ=\tfrac12[\cos(75^\circ-15^\circ)-\cos(75^\circ+15^\circ)]\) \(=\tfrac12[\cos 60^\circ-\cos 90^\circ]\) \(=\tfrac12\left[\tfrac12-0\right]\) \(=\tfrac14\) Final answer: \(\sin 75^\circ\sin 15^\circ=\tfrac14\).
Common mistakes to avoid
- Forgetting the factor of \(\tfrac12\) in front of the bracket
- Using \(\cos(A+B) - \cos(A-B)\) instead of \(\cos(A-B) - \cos(A+B)\)
- Replacing the product with a sum of sines instead of a difference of cosines
Did You Know?
Amplitude modulation, the AM in AM radio, is a direct application of multiplying two sine waves.
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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.