Question

Need help with this exercise it’s a review. More explanation

Answer 1

We have a right triangle, and we can apply the trigonometric ratios here.

We have the following situation:

The question is about finding the values of 9 and 12. To do that, we can proceed as follows:

1. We know that the cosine, sine, and tangent ratios are given by:

`\begin{gathered} \sin \theta=(opposite)/(hypotenuse) \\ \cos \theta=(adjacent)/(hypotenuse) \\ \tan \theta=(opposite)/(adjacent) \end{gathered}`

2. We have that the adjacent side to angle R is RS, and the opposite side to angle R is ST. Likewise, we have that the opposite side to angle T is RS and the adjacent side is ST.

3. We already know that:

`\begin{gathered} \cos R=(3)/(5)=(adjacent)/(hypotenuse) \\ (3)/(5)=(3)/(5)\cdot(3)/(3)=(9)/(15) \\ \cos R=(9)/(15) \end{gathered}`

We know that trigonometric ratios are expressed in their reduced form. However, we can see here that we have to multiply by 3/3 the cosine ratio to find the original length of the side. Therefore, the side RS = 9 (units). We can find the other side using the Pythagorean Theorem:

`\begin{gathered} ST^2+RS^2=15^2 \\ ST^2+9^2=15^2 \\ ST^2=15^2-9^2 \\ \sqrt[]{ST^2}=\sqrt[]{15^2-9^2} \\ ST=\sqrt[]{225-81} \\ ST=\sqrt[]{144} \\ ST=12 \end{gathered}`

4. Now, we got the necessary information to find the corresponding trigonometric ratios:

`\begin{gathered} \cos T=(12)/(15) \\ \sin T=(9)/(15)=(3)/(3)(3)/(5)=(3)/(5) \\ \sin R=(12)/(15)=(3)/(3)(4)/(5)=(4)/(5) \\ \cos R=(9)/(15)=(3)/(3)(3)/(5)=(3)/(5) \end{gathered}`

Therefore, we have:

A. Cos T = 12/15 (It is an option).

B. sinT = 3/5 (It is NOT an option sinT = 4/5).

C. sinR = 4/5 (It is an option).

D. cosR = 9/15 (It is NOT an option cosR = 12/15).

In summary, the answer choices that are possible trigonometric ratios for the triangle above are:

A. Cos T = 12/15

C. sinR = 4/5 (It is an option).