SOLUTION: Let theta be an angle in quadrant 3 such that the csc of theta = -6/5. Find the exact values of the tan of theta and the cos of theta.

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Question 754320: Let theta be an angle in quadrant 3 such that the csc of theta = -6/5.
Find the exact values of the tan of theta and the cos of theta.

Found 2 solutions by anteater, lwsshak3:
Answer by anteater(7) (Show Source):
You can put this solution on YOUR website!
You can start by drawing a right triangle in the third quadrant. They tell you that the cosecant of theta is -6/5. The cosecant is the ratio of the length of the hypotenuse to the length of the side opposite theta.
The hypotenuse always has a positive value, and you would label the hypotenuse "6" and the side opposite the angle "-5". You will then also want to find the adjacent side of your triangle (the side lying on the x-axis). Use the Pythagorean theorem to find this side.

You know a^2 + b^2 = c^2. For your triangle, then, (-5)^2 + b^2 = 6^2

25 + b^2 = 36 --> b^2 = 11 --> b = sqrt(11). And, since your triangle is in the 3rd quadrant, the adjacent side of your triangle will have a negative sign, so b = -sqrt(11)

So you have:
hypotenuse = 6
opposite = -5
adjacent = -sqrt(11)

You can use these values to find the tangent of theta and the cosine of theta.

Tan (theta) = opposite/adjacent = -5/-sqrt(11)

Since this has a radical expression in the denominator, they may want you to "rationalize" the denominator. To do that, you could multiply by sqrt(11)/sqrt(11):

(-5/-sqrt(11)) X (sqrt(11)/sqrt(11)) = 5sqrt(11)/11 This should be the tangent of your angle.

Cos (theta) = adjacent/hypotenuse = -sqrt(11)/6 This should be the cosine of your angle.

I hope this was helpful!

Answer by lwsshak3(11628) (Show Source):
You can put this solution on YOUR website!
Let theta be an angle in quadrant 3 such that the csc of theta = -6/5.
Find the exact values of the tan of theta and the cos of theta.
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use x for theta
csc x=-6/5
sin x=1/csc x=-5/6
cos x=-√1-sin^2(x)=-√(1-25/36)=-√(11/36)=-√11/6
tan x=sin x/cos x=-5/-√11=5√11/11