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Precalculus : Pre-Calculus

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Example Questions

Example Question #1 : Sum And Difference Identities For Cosine

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 #1 : Sum And Difference Identities

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

Possible Answers:

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

$-\frac{1}{2}$

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

$\frac{1}{2}$

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

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 #1 : Trigonometric Identities

Is the following equation an identity?

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

Possible Answers:

No it is not an identity

Yes it is an identity

The answer cannot be determined from the information provided.

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 #1 : Trigonometric Identities

Use trigonometric identities to solve the following equation for $\theta$ :

$tan^2\theta + sec^2\theta = 1$

Possible Answers:

$\theta = \pi/3+ n$

$\theta = \pi/4+ n\pi$

$\theta = \pi/2+ n\pi$

$\theta= n\pi$

$\theta = \pi/3+ n$

Correct answer:

$\theta= n\pi$

Explanation:

Use the trigonometric identities to switch sec into terms of tan:

$tan^2\theta + sec^2\theta = tan^2\theta + (tan^2 \theta + 1) = 2tan^2\theta + 1 = 1$

hence,

$2tan^2\theta = 0$

So we have $\tan \theta = 0$ , making $\theta= n \pi$

Therefore the solution is $\theta = n \pi$ for n being any integer.

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Example Question #2 : Solve Trigonometric Equations And Inequalities

Which of the following is not a solution to $\small 2sin^2(x)-3sin(x)=-1$ for $\small 0\leq x<2\pi$

Possible Answers:

$\small \frac{11\pi}{6}$

$\small \frac{5\pi}{6}$

$\small \frac{\pi}{6}$

$\small \frac{\pi}{2}$

Correct answer:

$\small \frac{11\pi}{6}$

Explanation:

We begin by setting the right side of the equation equal to 0.

$\small 2sin^2(x)-3sin(x)+1=0$

The equation might be easier to factor using the following substitution.

$\small w=sin(x)$

This gives the following

$\small 2w^2-3w+1=0$

This can be factored as follows

$\small \small (2w-1)(w-1)=0$

Therefore

$\small w=\frac{1}{2}\;or\;w=1$

Replacing our substitution therefore gives

$\small sin(x)=\frac{1}{2}\;or\;sin(x)=1$

Within our designated domain, we get three answers between our two equations.

$\small sin(x)=\frac{1}{2}$ when $\small x=\frac{\pi}{6}\;or\; \frac{5\pi}{6}$

$\small sin(x)=1$ when $\small x=\frac{\pi}{2}$

Therefore, the only choice that isn't correct is $\small \frac{11\pi}{6}$

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Example Question #3 : Solve Trigonometric Equations And Inequalities

Find one possible value of .

tan(5x)+17=-55

Possible Answers:

$17.84^{\circ}$

$0^{\circ}$

$446.02^{\circ}$

$-17.84^{\circ}$

$-446.02^{\circ}$

Correct answer:

$-17.84^{\circ}$

Explanation:

Begin by isolating the tangent side of the equation:

tan(5x)+17=-55

tan(5x)=-72

Next, take the inverse tangent of both sides:

$5x=tan^{-1} (-72)$

Divide by five to get the final answer:

$x=\frac{tan^{-1} (-72)}{5}=-17.84^{\circ}$

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Example Question #1 : Solve Trigonometric Equations And Inequalities

Use trigonometric identities to solve for the angle value.

$\tan^2\theta\sin^2\theta+\tan^2\theta\cos^2\theta+1=13$

Possible Answers:

$\theta\approx53.565$

$\theta\approx73.898$

$\theta\approx74.499$

$\theta\approx85.601$

Correct answer:

$\theta\approx73.898$

Explanation:

There are two ways to solve this problem. The first involves two trigonometric identities:

$\tan^2\theta\sin^2\theta+\tan^2\theta\cos^2\theta+1=13\\ \implies\tan^2\theta(\sin^2\theta+\cos^2\theta)+1=13\\ \implies\tan^2\theta+1=13\implies\sec^2\theta=13\\ \implies\sec\theta=\sqrt13\implies\cos\theta=\frac{\sqrt13}{13}\\ \implies \theta=\cos^{-1}\frac{\sqrt13}{13}}\approx73.898}$

The second method allows us to only use the first trigonometric identity:

$\tan^2\theta\sin^2\theta+\tan^2\theta\cos^2\theta+1=13\\ \implies\tan^2\theta(\sin^2\theta+\cos^2\theta)+1=13\\ \implies\tan^2\theta+1=13\implies\tan^2\theta=12\\ \implies\tan\theta=2\sqrt3\implies\theta=\tan^{-1}2\sqrt3\approx73.898}$

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Example Question #2 : Solve Trigonometric Equations And Inequalities

Use trigonometric identities to solve the equation for the angle value.

$4\cos^2\theta+\cos{2\theta}+\sin^2\theta=3$

Possible Answers:

$\theta=\frac{\pi}{6}$

$\theta\approx39.232$

$\theta=\frac{\pi}{2}$

$\theta\approx78.463$

Correct answer:

$\theta\approx39.232$

Explanation:

The simplest way to solve this problem is using the double angle identity for cosine. $\cos2\theta=\cos^2\theta-\sin^2\theta$

Substituting this value into the original equation gives us:

$4\cos^2\theta+\cos^2\theta-\sin^2\theta+\sin^2\theta=3\\ \implies5\cos^2\theta=3\implies\cos^2\theta=\frac{3}{5}\\ \cos\theta=\frac{\sqrt15}{5}\implies\theta=\cos^{-1}\frac{\sqrt15}{5}}\approx39.232$

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Example Question #213 : Pre Calculus

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

Possible Answers:

tan(x)

csc^2(x)

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

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

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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Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Sum And Difference Identities For Cosine

Example Question #1 : Sum And Difference Identities

Example Question #1 : Trigonometric Identities

Example Question #9 : Sum And Difference Identities

Example Question #1 : Trigonometric Identities

Example Question #2 : Solve Trigonometric Equations And Inequalities

Example Question #3 : Solve Trigonometric Equations And Inequalities

Example Question #1 : Solve Trigonometric Equations And Inequalities

Example Question #2 : Solve Trigonometric Equations And Inequalities

Example Question #213 : Pre Calculus

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