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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.
