In a right triangle...

Question

In a right triangle, tan θ= -5/12 and π/2 < θ < π. Which of the following is true?

 

 

a) sec θ= -13/12

 

b) sin θ= 5/13

 

c) csc θ= 13/5

 

d) cot θ= 12/5

Answer 1

The problem gives you a couple useful pieces of information. We'll use the fact that tan(θ) = -5/12 to figure out the side-lengths of the triangle θ is a part of. That will tell us the values of sin(θ), cos(θ), sec(θ), csc(θ), and cot(θ), but not whether they are positive or negative. We'll use the fact that π/2 < θ < π to determine whether each value is positive or negative.

 

 

 

First, tan(θ) = -5/12. Since tan = opposite/adjacent, θ's opposite side has length 5 and its adjacent side has length 12. Plug those values into the Pythagorean Theorem to solve for the triangle's hypotenuse:

 

c² = a² + b²

c² = 5² + 12²

c² = 25 + 144

c² = 169

c = 13

 

So, opposite = 5, adjacent = 12, and hypotenuse = 13.

 

 

Next, π/2 < θ < π. That means that we're in Quadrant II. In Quadrant II, sine and cosecant are positive, while cosine, tangent, secant, and cotangent are negative.

 

 

 

Put all of that together to get the values of every trig function for θ:

 

sin(θ) = +opposite/hypotenuse = 5/13

cos(θ) = -adjacent/hypotenuse = -12/13

tan(θ) = -opposite/adjacent = -5/12

csc(θ) = 1/sin(θ) = +hypotenuse/opposite = 13/5

sec(θ) = 1/cos(θ) = -hypotenuse/adjacent = -13/12

cot(θ) = 1/tan(θ) = -adjacent/opposite = -12/5