Answer:
- csc(θ) = -5/4
- cos(θ) = 3/5
Step-by-step explanation:
In quadrant IV, the tangent is negative, so the first answer choice cannot be correct.
The triangle involved is a 3-4-5 triangle, so the second answer choice cannot be correct.
The cosecant can be found several ways. One is to find the sine of the angle (-4/5 — compare to answer choice 2), then invert it. (csc(θ) = 1/sin(θ)) Here that means csc(θ) = 1/(-4/5) = -5/4, in agreement with the third answer choice.
cos(θ) = 1/sec(θ) = 1/(5/3) = 3/5, in agreement with the fourth answer choice.
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<em>Comment on cosecant</em>
You know from SOH CAH TOA that ...
Cos = Adjacent/Hypotenuse
Then Sec = 1/Cos = Hypotenuse/Adjacent. This can help you draw the triangle, which you quickly realize is a 3-4-5 right triangle. If not, you can use the Pythagorean theorem to find the opposite side length as ...
√(5²-3²) = √16 = 4
Remember the triangle is being drawn in the 4th quadrant, so the side opposite the angle will be negative.
Of course, Csc = 1/Sin = Hypotenuse/Opposite, so is 5/(-4) = -5/4.
