Express products as sums.
Express sums as products.
Expressing Products as Sums
We have already learned a number of formulas useful for expanding or simplifying trigonometric expressions, but sometimes we may need to express the product of cosine and sine as a sum. We can use the product-to-sum formulas , which express products of trigonometric functions as sums. Let’s investigate the cosine identity first and then the sine identity.
Expressing Products as Sums for Cosine
We can derive the product-to-sum formula from the sum and difference identities for cosine . If we add the two equations, we get:
[latex]\begin{gathered}\cos \alpha \cos \beta +\sin \alpha \sin \beta =\cos \left(\alpha -\beta \right)\\\underline{ +\cos \alpha \cos \beta -\sin \alpha \sin \beta =\cos \left(\alpha +\beta \right)} \\ 2\cos \alpha \cos \beta =\cos \left(\alpha -\beta \right)+\cos \left(\alpha +\beta \right)\end{gathered}[/latex]
Then, we divide by [latex]2[/latex] to isolate the product of cosines:
[latex]\cos \alpha \cos \beta =\frac{1}{2}\left[\cos \left(\alpha -\beta \right)+\cos \left(\alpha +\beta \right)\right][/latex]
product-to-sum formulas
The product-to-sum formulas are as follows:
[latex]\cos \alpha \cos \beta =\frac{1}{2}\left[\cos \left(\alpha -\beta \right)+\cos \left(\alpha +\beta \right)\right][/latex]
[latex]\sin \alpha \cos \beta =\frac{1}{2}\left[\sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)\right][/latex]
[latex]\sin \alpha \sin \beta =\frac{1}{2}\left[\cos \left(\alpha -\beta \right)-\cos \left(\alpha +\beta \right)\right][/latex]
[latex]\cos \alpha \sin \beta =\frac{1}{2}\left[\sin \left(\alpha +\beta \right)-\sin \left(\alpha -\beta \right)\right][/latex]
How To: Given a product of cosines, express as a sum.
Write the formula for the product of cosines.
Substitute the given angles into the formula.
Simplify.
Write the following product of cosines as a sum: [latex]2\cos \left(\frac{7x}{2}\right)\cos \frac{3x}{2}[/latex].
Show Solution
We begin by writing the formula for the product of cosines:
[latex]\cos \alpha \cos \beta =\frac{1}{2}\left[\cos \left(\alpha -\beta \right)+\cos \left(\alpha +\beta \right)\right][/latex]
We can then substitute the given angles into the formula and simplify.
[latex]\begin{align}2\cos \left(\frac{7x}{2}\right)\cos \left(\frac{3x}{2}\right)&=\left(2\right)\left(\frac{1}{2}\right)\left[\cos \left(\frac{7x}{2}-\frac{3x}{2}\right)+\cos \left(\frac{7x}{2}+\frac{3x}{2}\right)\right] \\ &=\left[\cos \left(\frac{4x}{2}\right)+\cos \left(\frac{10x}{2}\right)\right] \\ &=\cos 2x+\cos 5x \end{align}[/latex]
Use the product-to-sum formula to write the product as a sum or difference: [latex]\cos \left(2\theta \right)\cos \left(4\theta \right)[/latex].
Show Solution
[latex]\frac{1}{2}\left(\cos 6\theta +\cos 2\theta \right)[/latex]
Expressing the Product of Sine and Cosine as a Sum
Next, we will derive the product-to-sum formula for sine and cosine from the sum and difference formulas for sine . If we add the sum and difference identities, we get:
[latex]\begin{gathered}\sin \left(\alpha +\beta \right)=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\\underline{ +\text{ }\sin \left(\alpha -\beta \right)=\sin \alpha \cos \beta -\cos \alpha \sin \beta}\\ \sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)=2\sin \alpha \cos \beta \end{gathered}[/latex]
Then, we divide by 2 to isolate the product of cosine and sine:
[latex]\sin \alpha \cos \beta =\frac{1}{2}\left[\sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)\right][/latex]
Express the following product as a sum containing only sine or cosine and no products: [latex]\sin \left(4\theta \right)\cos \left(2\theta \right)[/latex].
Show Solution
Write the formula for the product of sine and cosine. Then substitute the given values into the formula and simplify.
[latex]\begin{align}\sin \alpha \cos \beta &=\frac{1}{2}\left[\sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)\right] \\ \sin \left(4\theta \right)\cos \left(2\theta \right)&=\frac{1}{2}\left[\sin \left(4\theta +2\theta \right)+\sin \left(4\theta -2\theta \right)\right] \\ &=\frac{1}{2}\left[\sin \left(6\theta \right)+\sin \left(2\theta \right)\right] \end{align}[/latex]
Use the product-to-sum formula to write the product as a sum: [latex]\sin \left(x+y\right)\cos \left(x-y\right)[/latex].
Show Solution
[latex]\frac{1}{2}\left(\sin 2x+\sin 2y\right)\\[/latex]
Expressing Products of Sines in Terms of Cosine
Expressing the product of sines in terms of cosine is also derived from the sum and difference identities for cosine. In this case, we will first subtract the two cosine formulas:
[latex]\begin{gathered}\cos \left(\alpha -\beta \right)=\cos \alpha \cos \beta +\sin \alpha \sin \beta \\ \underline{ -\text{ }\cos \left(\alpha +\beta \right)=-\left(\cos \alpha \cos \beta -\sin \alpha \sin \beta \right)} \\ \cos \left(\alpha -\beta \right)-\cos \left(\alpha +\beta \right)=2\sin \alpha \sin \beta \end{gathered}[/latex]
Then, we divide by 2 to isolate the product of sines:
[latex]\sin \alpha \sin \beta =\frac{1}{2}\left[\cos \left(\alpha -\beta \right)-\cos \left(\alpha +\beta \right)\right][/latex]
Similarly we could express the product of cosines in terms of sine or derive other product-to-sum formulas.
Write [latex]\cos \left(3\theta \right)\cos \left(5\theta \right)[/latex] as a sum or difference.
Show Solution
We have the product of cosines, so we begin by writing the related formula. Then we substitute the given angles and simplify.
[latex]\begin{align}\cos \alpha \cos \beta &=\frac{1}{2}\left[\cos \left(\alpha -\beta \right)+\cos \left(\alpha +\beta \right)\right] \\ \cos \left(3\theta \right)\cos \left(5\theta \right)&=\frac{1}{2}\left[\cos \left(3\theta -5\theta \right)+\cos \left(3\theta +5\theta \right)\right] \\ &=\frac{1}{2}\left[\cos \left(2\theta \right)+\cos \left(8\theta \right)\right] && \text{Use even-odd identity}. \end{align}[/latex]
Use the product-to-sum formula to evaluate [latex]\cos \frac{11\pi }{12}\cos \frac{\pi }{12}[/latex].
Show Solution
[latex]\frac{-2-\sqrt{3}}{4}[/latex]