Question

I don’t understand how to do this

Answer 1

Step-by-step explanation:

θ is in quadrant IV, so:

sin θ < 0

cos θ > 0

tan θ = sin θ / cos θ < 0

csc θ = 1 / sin θ < 0

sec θ = 1 / cos θ > 0

Without doing any calculations, we can see only the third option fits (in the second option, sin θ / cos θ = -9/18, not -18/9.  In the fourth option, csc θ and sec θ are switched).

Let's go ahead and calculate the values.  There are several ways to solve this.  One way is to use Pythagorean identities (ex., 1 + cot²θ = csc²θ).  Another way is to simply draw a triangle in the fourth quadrant.

cot θ = 1 / tan θ, and tan θ = opposite / adjacent.  So cot θ = adjacent / opposite.  If we draw a triangle with angle θ, where the adjacent side is 9 and the opposite side is -18, then we can use Pythagorean theorem to find the hypotenuse:

c² = a² + b²

c² = (9)² + (-18)²

c = √405

Therefore:

sin θ = -18 / √405

cos θ = 9 / √405

csc θ = √405 / 18

sec θ = √405 / 9

tan θ = -18/9