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Precalculus : Sum and Difference Identities

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Example Question #1 : Sum And Difference Identities

According to the trigonometric identities, 1+cot^2(x)=?

Possible Answers:

$\frac{1}{tan(x)}$

csc^2(x)

tan(x)

$\frac{sin(x)}{cos(x)}$

Correct answer:

csc^2(x)

Explanation:

The trigonometric identity 1+cot^2(x)=csc^2(x) , is an important identity to memorize.

Some other identities that are important to know are:

$\frac{sin(x)}{cos(x)}=tan(x)$

$cot(x)=\frac{1}{tan(x)}$

sin^2(x)+cos^2(x)=1

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Example Question #2 : Sum And Difference Identities

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

Possible Answers:

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

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

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

$\frac{1}{2}$

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

Correct answer:

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

Explanation:

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

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Example Question #1 : Sum And Difference Identities

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

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:

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

and so this is again not an identity

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Example Question #4 : Sum And Difference Identities

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

Possible Answers:

The question cannot be answered based on the information provided.

No it is not an identity

Yes it is an identity

Correct answer:

No it is not an identity

Explanation:

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

Therefore, this is not an identity.

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Example Question #5 : Sum And Difference Identities

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

Possible Answers:

$\frac{-1}{2}$

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

$\frac{2\sqrt{2}}{4}$

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

$\frac{1}{2}$

Correct answer:

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

Explanation:

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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Example Question #1 : Sum And Difference Identities

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

Possible Answers:

No it is not an identity.

Yes it is an identity.

It cannot be determined from the given information.

Correct answer:

No it is not an identity.

Explanation:

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

and due to this inequality, this is not an identity

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Example Question #7 : Sum And Difference Identities

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

Possible Answers:

$\frac{2\sqrt{2}}{4}$

$-\frac{1}{2}$

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

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

$\frac{1}{2}$

Correct answer:

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

Explanation:

Using the identity, we can break up the 255 into 300-45 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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Example Question #8 : Sum And Difference Identities

Is the following equation an identity?

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

Possible Answers:

Yes it is an identity

The answer cannot be determined from the information provided.

No it is not an identity

Correct answer:

No it is not an identity

Explanation:

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

Due to this inequality, this is not an identity.

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Example Question #9 : Sum And Difference Identities

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

Possible Answers:

$\frac{1}{2}$

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

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

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

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

Correct answer:

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

Explanation:

Here we break up the 165 into 135+30 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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Example Question #10 : Sum And Difference Identities

Evaluate the exact value of:

2cos(75)

Possible Answers:

$\frac{\sqrt6-\sqrt2}{4}$

$\frac{\sqrt2-\sqrt6}{4}$

$\frac{\sqrt6+\sqrt2}{2}$

$\frac{\sqrt6+\sqrt2}{4}$

$\frac{\sqrt6-\sqrt2}{2}$

Correct answer:

$\frac{\sqrt6-\sqrt2}{2}$

Explanation:

In order to solve 2cos(75) , two special angles will need to be used to solve for the exact values.

The angles chosen are A=30 and B=45 degrees, since:

A+B=30+45 =75

Write the formula for the cosine additive identity.

cos(A+B)= cosAcosB-sinAsinB

Substitute the known variables.

$cos(30)cos(45)-sin(30)sin(45)=\left(\frac{\sqrt3}{2}\right)\left(\frac{\sqrt2}{2}\right)-\left(\frac{1}{2}\right)\left(\frac{\sqrt2}{2}\right)$

$=\frac{\sqrt6}{4}-\frac{\sqrt2}{4}=\frac{\sqrt6-\sqrt2}{4}$

$2cos(75)=2\left(\frac{\sqrt6-\sqrt2}{4}\right)=\frac{\sqrt6-\sqrt2}{2}$

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Precalculus : Sum and Difference Identities

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