Question

30 Points please help.

Answer 1

2 Answers: Choice C and Choice E

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Explanation:

θ = greek letter theta = reference angle

Using the unit circle, you should find that when θ is pi/3, we have

• cos(θ) = cos(pi/3) = 1/2
• sin(θ) = sin(pi/3) = sqrt(3)/2

Dividing sine over cosine gets us tangent

tan(θ) = sin(θ)/cos(θ) = sqrt(3)/2 divide over (1/2) = sqrt(3)

Effectively, the denominators '2' cancel out when dividing the two fractions. The result we get here is not sqrt(3)/2, so we can rule out choice A.

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Choice B can be ruled out because

cos(0) = 1

sin(0) = 0

So,

tan(0) = sin(0)/cos(0) = 0/1 = 0

which doesn't match with y = pi

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Following the same ideas as mentioned before:

cos(pi/4) = sqrt(2)/2

sin(pi/4) = sqrt(2)/2 ... it's not a typo, sine and cosine are the same here

tan(pi/4) = 1 after dividing the two items above

We end up with y = 1 as the screenshot shows, so (pi/4, 1) is one point on the graph of y = tan(x).

Choice C is one of the answers

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Choice D however is not one of the answers because

sin(pi/2) = 1

cos(pi/2) = 0

tan(pi/2) = undefined, because the denominator cosine is 0 in this case

So there's a vertical asymptote at x = pi/2 for y = tan(x)

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Choice E is another answer, because,

sin(pi) = 0

cos(pi) = -1

tan(pi) = sin(pi)/cos(pi) = 0/(-1) = 0

This shows (pi, 0) is a point on y = tan(x).

The graph is shown below. Points C and E are on the blue tangent curve, while everything else isn't.

I used GeoGebra to create the graph. Desmos is also a handy tool that can perform similar tasks.