Q: Which of the following expressions is equivalent to $$\cos\theta$$?

Use the cofunction identity: $$sinleft( heta+rac{pi}{2} ight)=sin hetacosrac{pi}{2}+cos hetasinrac{pi}{2}.$$ Since: $$cosrac{pi}{2}=0,$$ $$sinrac{pi}{2}=1,$$ we get: $$sinleft( heta+rac{pi}{2} ight)=0+cos heta=cos heta.$$ Final answer: $$sinleft( heta+rac{pi}{2} ight)$$

Q: If $$f(x)=\cos(\sin^{-1}x)$$, what is the value of $$f\left(\frac{3}{5}\right)$$?

Let: $$ heta=sin^{-1}left(rac{3}{5} ight).$$ Then: $$sin heta=rac{3}{5}.$$ Since $$ heta$$ is in the principal range of inverse sine, $$ heta$$ is in Quadrant I or IV. For $$sin^{-1}left(rac{3}{5} ight)$$, the principal angle is acute, so use a right triangle: Opposite $$=3$$ Hypotenuse $$=5$$ Adjacent $$=sqrt{5^2-3^2}=sqrt{25-9}=4$$ Therefore: $$cos heta=rac{4}{5}.$$ Final answer: $$rac{4}{5}$$

Q: If the terminal side of angle $$A$$ in standard position is in Quadrant I and $$\sin A=\frac{12}{13}$$, what is the value of $$\cos A\cdot\tan A$$?

Use the identity: $$ an A=rac{sin A}{cos A}.$$ Then: $$cos Acdot an A=cos Acdotrac{sin A}{cos A}.$$ Simplify: $$cos Acdot an A=sin A.$$ Substitute the given value: $$sin A=rac{12}{13}.$$ Final answer: $$rac{12}{13}$$

Question 1

Which of the following expressions is equivalent to $$\cos	heta$$?

Accepted Answer

Use the cofunction identity:
$$sinleft(	heta+rac{pi}{2}ight)=sin	hetacosrac{pi}{2}+cos	hetasinrac{pi}{2}.$$

Since:
$$cosrac{pi}{2}=0,$$
$$sinrac{pi}{2}=1,$$

we get:
$$sinleft(	heta+rac{pi}{2}ight)=0+cos	heta=cos	heta.$$

Final answer:
$$sinleft(	heta+rac{pi}{2}ight)$$

Question 2

At which value of $$x$$ does the graph of $$f(x)=	an\left(x+\frac{\pi}{6}ight)$$ have a vertical asymptote?

Accepted Answer

A vertical asymptote for tangent occurs when its input equals:
$$rac{pi}{2}+kpi.$$

Set the inside equal to $$rac{pi}{2}$$:
$$x+rac{pi}{6}=rac{pi}{2}.$$

Solve for $$x$$:
$$x=rac{pi}{2}-rac{pi}{6}=rac{3pi}{6}-rac{pi}{6}=rac{2pi}{6}=rac{pi}{3}.$$

Final answer:
$$rac{pi}{3}$$

Question 3

If $$f(x)=\cos(\sin^{-1}x)$$, what is the value of $$f\left(\frac{3}{5}\right)$$?

Accepted Answer

Let:
$$	heta=sin^{-1}left(rac{3}{5}ight).$$

Then:
$$sin	heta=rac{3}{5}.$$

Since $$	heta$$ is in the principal range of inverse sine, $$	heta$$ is in Quadrant I or IV. For $$sin^{-1}left(rac{3}{5}ight)$$, the principal angle is acute, so use a right triangle:
Opposite $$=3$$
Hypotenuse $$=5$$
Adjacent $$=sqrt{5^2-3^2}=sqrt{25-9}=4$$

Therefore:
$$cos	heta=rac{4}{5}.$$

Final answer:
$$rac{4}{5}$$

Question 4

If $$\sin x=\frac{5}{13}$$, $$\cos y=\frac{12}{13}$$, and $$x$$ and $$y$$ are acute angles, what is the value of $$\cos(x-y)$$?

Accepted Answer

Use the identity:
$$cos(x-y)=cos xcos y+sin xsin y.$$

First find the missing trig values.
Since $$sin x=rac{5}{13}$$ and $$x$$ is acute:
$$cos x=rac{12}{13}.$$

Since $$cos y=rac{12}{13}$$ and $$y$$ is acute:
$$sin y=rac{5}{13}.$$

Substitute into the identity:
$$cos(x-y)=left(rac{12}{13}ight)left(rac{12}{13}ight)+left(rac{5}{13}ight)left(rac{5}{13}ight).$$

Simplify:
$$cos(x-y)=rac{144}{169}+rac{25}{169}=rac{169}{169}=1.$$

Final answer:
$$1$$

Question 5

Which of the following choices is equivalent to the expression $$\sin	heta(\csc	heta-\sin	heta)$$?

Accepted Answer

Start with the expression:
$$sin	heta(csc	heta-sin	heta).$$

Use the identity:
$$csc	heta=rac{1}{sin	heta}.$$

Substitute:
$$sin	hetaleft(rac{1}{sin	heta}-sin	hetaight).$$

Distribute $$sin	heta$$:
$$1-sin^2	heta.$$

Use the Pythagorean identity:
$$1-sin^2	heta=cos^2	heta.$$

Final answer:
$$cos^2	heta$$

Question 6

If the terminal side of angle $$A$$ in standard position is in Quadrant I and $$\sin A=\frac{12}{13}$$, what is the value of $$\cos A\cdot	an A$$?

Accepted Answer

Use the identity:
$$	an A=rac{sin A}{cos A}.$$

Then:
$$cos Acdot	an A=cos Acdotrac{sin A}{cos A}.$$

Simplify:
$$cos Acdot	an A=sin A.$$

Substitute the given value:
$$sin A=rac{12}{13}.$$

Final answer:
$$rac{12}{13}$$

Question 7

If $$x$$ is a positive acute angle and $$\sin x=a$$, which of the following is an expression for $$	an x$$ in terms of $$a$$?

Accepted Answer

Start with:
$$sin x=a.$$

Since $$x$$ is acute:
$$cos x=sqrt{1-sin^2x}=sqrt{1-a^2}.$$

Use the tangent identity:
$$	an x=rac{sin x}{cos x}.$$

Substitute:
$$	an x=rac{a}{sqrt{1-a^2}}.$$

Final answer:
$$rac{a}{sqrt{1-a^2}}$$

Question 8

If $$	an x=1$$ and $$\cos x=-\frac{\sqrt{2}}{2}$$, find the measure of $$\angle x$$.

Accepted Answer

First use:
$$	an x=1.$$

Reference angles where tangent equals $$1$$ are:
$$rac{pi}{4}$$ and angles coterminal with it.

Now use:
$$cos x=-rac{sqrt{2}}{2}.$$

Cosine is negative in Quadrant II and Quadrant III.
Tangent is positive in Quadrant I and Quadrant III.

The only quadrant that satisfies both conditions is Quadrant III.

So the angle is:
$$x=pi+rac{pi}{4}=rac{5pi}{4}.$$

Final answer:
$$rac{5pi}{4}$$

Quiz 1

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