Question

∠θ is a third-quadrant angle and tan θ = 5/12 ; using this information find the value. csc θ =

a. 13/12
b. 13/5
c. -13/5

Answer 1

To find csc θ when tan θ is 5/12 and θ is a third-quadrant angle, we calculate the hypotenuse of a right triangle using the Pythagorean theorem and then find the sine of θ. As sine is negative in the third quadrant, csc θ is the negative reciprocal of sine, yielding the value of -13/5.

Step-by-step explanation:

To find the value of csc θ, we need to solve the mathematical problem completely by understanding the properties of trigonometric functions in different quadrants. Given that tan θ = 5/12 and θ is a third-quadrant angle, where both sine and cosine are negative, we can use this information to find sine (θ), which is needed to calculate csc (θ).

In a right triangle, the tangent (tan) of an angle is the ratio of the opposite side to the adjacent side. Here, tan θ equals 5/12, which means the opposite side is 5 units and the adjacent side is 12 units. Because we are dealing with a third-quadrant angle, we must also consider that both components are negative in this quadrant.

To find the hypotenuse, we use the Pythagorean theorem:

1. The opposite side is -5 (negative because θ is in the third quadrant).
2. The adjacent side is -12 (negative for the same reason).
3. Calculate the hypotenuse using the Pythagorean theorem: (-5)² + (-12)² = h², which simplifies to 25 + 144 = h², and finally, h = sqrt(169) = 13.
4. Because θ is in the third quadrant, the hypotenuse is always positive, so h = 13.
5. The sine (sin) of θ is the opposite over hypotenuse, so sin θ = -5/13 (negative because θ is in the third quadrant).
6. The cosecant (csc) is the reciprocal of the sine, hence csc θ = -1/(sin θ) = -13/5.

The correct answer, given the options, is c. -13/5.