Sum and Difference Identities For Cosine - Pre-Calculus

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Question

Use the sum or difference identity to find the exact value: $cos255^\circ$

Answer

Using the identity, we can break up the 255 into and then solve: $cos300\cdot cos45+sin300\cdot sin45\rightarrow \frac{1}{2}\cdot \frac{\sqrt{2}}{2}+\frac{-\sqrt{3}}{2}\cdot \frac{\sqrt{2}}{2}\rightarrow \frac{\sqrt{2}-\sqrt{6}}{4}$

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Question

Is the following equation an identity? $cos(x-\pi)=cos(x)$

Answer

$cos(x-\pi)=cos(x)\rightarrow cosxcos\pi+sinxsin\pi=cosx\rightarrow cosx(-1)+sinx(0)=cosx\rightarrow -cosx eq cosx$

and due to this inequality, this is not an identity

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Question

Use the sum or difference identity to find the exact value: $\cos 255^{\circ}$

Answer

Using the identity, we can break up the into and then solve: $\cos 300\cdot \cos 45+\sin 300\cdot \sin 45\rightarrow \frac{1}{2}\cdot \frac{\sqrt{2}}{2}+\frac{-\sqrt{3}}{2}\cdot \frac{\sqrt{2}}{2}\rightarrow \frac{\sqrt{2}-\sqrt{6}}{4}$ and so the correct answer is $\frac{\sqrt{2}-\sqrt{6}}{4}$ .

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Question

Use the sum or difference identity to find the exact value of $\sin 165^{\circ}$ .

Answer

Here we break up the into and solve using the sin identity: $\sin 135\cdot \cos 30+\cos 135\cdot \sin 30\rightarrow \frac{\sqrt{2}}2{}\cdot \frac{\sqrt{3}}2{}+\frac{-\sqrt{2}}2{}\cdot \frac{1}{2}\rightarrow \frac{\sqrt{6}-\sqrt{2}}{4}$ and so here the credited answer is $\frac{\sqrt{6}-\sqrt{2}}{4}$ .

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Question

Is the following equation an identity?

$\cos \left ( x-\pi \right )=\cos \left ( x \right )$

Answer

$\cos \left ( x-\pi \right )=\cos \left ( x \right )\rightarrow \cos x\cos \pi +\sin x\sin \pi =\cos x\rightarrow \cos x(-1)+\sin x(0)=\cos x\rightarrow -\cos x eq \cos x$

Due to this inequality, this is not an identity.

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