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Precalculus : Sum and Difference Identities For Sine

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Example Question #1 : Sum And Difference Identities

Use the sum or difference identity to find the exact value: $sin165^\circ$ $\displaystyle sin165^\circ$

Possible Answers:

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

$\frac{\sqrt{6}-\sqrt{2}}{4}$ $\displaystyle \frac{\sqrt{6}-\sqrt{2}}{4}$

$\frac{\sqrt{2}-\sqrt{6}}{4}$ $\displaystyle \frac{\sqrt{2}-\sqrt{6}}{4}$

$\frac{\sqrt{6}+\sqrt{2}}{4}$ $\displaystyle \frac{\sqrt{6}+\sqrt{2}}{4}$

$\frac{\sqrt{6}}{4}$ $\displaystyle \frac{\sqrt{6}}{4}$

Correct answer:

$\frac{\sqrt{6}-\sqrt{2}}{4}$ $\displaystyle \frac{\sqrt{6}-\sqrt{2}}{4}$

Explanation:

Again, here we break up the 165 into 135+30 $\displaystyle 135+30$ and solve using the sin identity: $sin135\cdot cos30+cos135\cdot sin30\rightarrow \frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}+\frac{-\sqrt{2}}{2}\cdot \frac{1}{2}\rightarrow \frac{\sqrt{6}-\sqrt{2}}{4}$ $\displaystyle sin135\cdot cos30+cos135\cdot sin30\rightarrow \frac{\sqrt{2}}{2}\cdot \frac{\sqrt{3}}{2}+\frac{-\sqrt{2}}{2}\cdot \frac{1}{2}\rightarrow \frac{\sqrt{6}-\sqrt{2}}{4}$

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Example Question #1 : Sum And Difference Identities

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

Possible Answers:

It cannot be determined from the given information.

Yes it is an identity.

No it is not an identity.

Correct answer:

No it is not an identity.

Explanation:

$sin(x-\pi)=sinx\rightarrow sinxcos\pi-cosxsin\pi=sinx\rightarrow sinx\cdot (-1)-cosx\cdot (0)\rightarrow -sinx-0=sinx\rightarrow -sinx\neq sinx$ $\displaystyle sin(x-\pi)=sinx\rightarrow sinxcos\pi-cosxsin\pi=sinx\rightarrow sinx\cdot (-1)-cosx\cdot (0)\rightarrow -sinx-0=sinx\rightarrow -sinx\neq sinx$

and so this is again not an identity

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Example Question #1 : Trigonometric Identities

Is the following equation an identity? $\sin \left ( x-\pi \right )=\sin \left ( x \right )$ $\displaystyle \sin \left ( x-\pi \right )=\sin \left ( x \right )$ ?

Possible Answers:

Yes it is an identity

No it is not an identity

The question cannot be answered based on the information provided.

Correct answer:

No it is not an identity

Explanation:

$\sin \left ( x-\pi \right )=\sin x\rightarrow \sin x\cos \pi -\cos x\sin \pi = \sin x\rightarrow \sin x\cdot \left ( -1 \right )-\cos x\cdot \left ( 0 \right )\rightarrow -\sin x-0=\sin x\neq \sin x$ $\displaystyle \sin \left ( x-\pi \right )=\sin x\rightarrow \sin x\cos \pi -\cos x\sin \pi = \sin x\rightarrow \sin x\cdot \left ( -1 \right )-\cos x\cdot \left ( 0 \right )\rightarrow -\sin x-0=\sin x\neq \sin x$

Therefore, this is not an identity.

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Precalculus : Sum and Difference Identities For Sine

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Sum And Difference Identities

Example Question #1 : Sum And Difference Identities

Example Question #1 : Trigonometric Identities

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