Triple-Angle Formulas and Linear Combinations

Combination of the sum and double angle formulas; set of terms added or subtracted with a constant multiplier.

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Sum to Product Formulas for Cosine and Sine

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sum to product formula relates the sum or difference of two trigonometric functions to the product of two trigonometric functions.

sinαsinβ=2sinαβ2×cosα+β2cosα+cosβ=2cosα+β2×cosαβ2cosαcosβ=2sinα+β2×sinαβ2\begin{align*} \sin \alpha - \sin \beta = 2 \sin \frac{\alpha - \beta}{2} \times \cos \frac{\alpha + \beta}{2}\\ \cos \alpha + \cos \beta = 2 \cos \frac{\alpha + \beta}{2} \times \cos \frac{\alpha - \beta}{2}\\ \cos \alpha - \cos \beta = -2 \sin \frac{\alpha + \beta}{2} \times \sin \frac{\alpha - \beta}{2}\\\end{align*}

This is found by adding two trigonometric functions and using identities to find the product of them.

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Product to Sum

product to sum formula relates the product of two trigonometric functions to the sum of two trigonometric functions.

We need to start with the cosine of the difference of two angles to find the product of sine of the two angles.

cos(ab)=cosacosb+sinasinb and cos(a+b)=cosacosbsinasinbcos(ab)cos(a+b)=cosacosb+sinasinb(cosacosbsinasinb)cos(ab)cos(a+b)=cosacosb+sinasinbcosacosb+sinasinbcos(ab)cos(a+b)=2sinasinb12[cos(ab)cos(a+b)]=sinasinb\begin{align*}& \cos(a - b) = \cos a \cos b + \sin a \sin b \ \text {and} \ \cos(a + b) = \cos a \cos b - \sin a \sin b \\ & \cos(a - b) - \cos (a + b) = \cos a \cos b + \sin a \sin b - (\cos a \cos b - \sin a \sin b)\\ & \cos(a - b) - \cos (a + b) = \cos a \cos b + \sin a \sin b - \cos a \cos b + \sin a \sin b\\ & \qquad \qquad \qquad \qquad \cos (a - b) - \cos (a + b) = 2 \sin a \sin b\\ & \qquad \qquad \qquad \qquad \frac{1}{2}\left[ \cos (a - b) - \cos (a+ b)\right] = \sin a \sin b\end{align*}

Hint: Derive the products so that you can get the full product to sum formulas.

cosαcosβ=12[cos(αβ)+cos(α+β)]sinαcosβ=12[sin(α+β)+sin(αβ)]cosαsinβ=12[sin(α+β)sin(αβ)]\begin{align*}\cos \alpha \cos \beta = \frac {1}{2} \left [ \cos (\alpha - \beta) + \cos (\alpha + \beta) \right ]\\ \sin \alpha \cos \beta = \frac {1}{2} \left [\sin (\alpha + \beta) + \sin (\alpha - \beta) \right ]\\ \cos \alpha \sin \beta = \frac {1}{2} \left[\sin (\alpha + \beta) - \sin (\alpha - \beta) \right ]\end{align*}

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Triple-Angle Formulas

linear combination is a set of terms that are added or subtracted from each other with a multiplicative constant in front of each term.

A triple angle identity relates the a trigonometric function of three times an argument to a set of trigonometric functions, each containing the original argument.

Hint: You have to combine the sum formula and double angle formula to create the triple-angle formulas.

Acosx+Bsinx=Ccos(xD)\begin{align*}A \cos x + B \sin x = C \cos(x - D)\end{align*} , where C=A2+B2\begin{align*}C = \sqrt{A^2 + B^2}\end{align*} , cosD=AC\begin{align*}\cos D = \frac{A}{C} \end{align*} and sinD=BC\begin{align*}\sin D = \frac{B}{C}\end{align*} .

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