Precalculus

Question

Use the theorem below to answer the question.
Suppose θ is an acute angle residing in a right triangle. If the length of the side adjacent to θ is a, the length of the side opposite θ is b, and the length of the hypotenuse is c, then
cos(θ) = a/c and sin(θ) = b/c .If θ = 52.5° and the side opposite θ has length 309, how long is the side adjacent to θ? (Round your answer to two decimal places.)

Answer 1

Remember that sin(θ)=oposite/hypotenuse, cos(θ)=adjacent/hypotenuse.

 

Since we know θ, we know the opposite side length, and we want to know the adjacent side length, we want to use tangent. However, we need to use the identities above.

Tangent can be defined as sin(θ)/cos(θ).

 

This will give you tan(θ)=(b/c)/(a/c), which will simplify to tan(θ)=b/a. Since we want to find a, we can solve for it, giving us the equation:

 

b=a*tan(θ)

 

Plug in your knowns into that equation, and you will get the length of the adjacent side (b).

 

Alternatively, we can find the length of the hypotenuse (c) using sine (sin(θ)=a/c => c=a/sin(θ)), then use cosine along with our calculated (c) to get the adjacent side, (b). 

 

cos(θ)=b/c => b=cos(θ)*c

 

Plugging in our "c" we found using sine along with the value of θ will give you your answer.