Chapter 4.3, Problem 92E

(a)

To determine

To find: The side opposite to $\theta$ in the given triangle.

Answer to Problem 92E

The side opposite to the angle $\theta$ is $y$ .

Explanation of Solution

Given Information:

The given triangle is as follows.

  

Figure (1)

Calculation:

The side opposite to the right angle is called hypotenuse. It is the largest side of a right-angled triangle.

In the given triangle, the side opposite to the angle $\theta$ is $y$ .

Therefore, the side opposite to the angle $\theta$ is $y$ .

(b)

To determine

To find:The side adjacent to $90-\theta$ .

Answer to Problem 92E

The side adjacent to angle $90-\theta$ is $y$ .

Explanation of Solution

Given Information:

The given triangle is as follows.

  

Calculation:

The side opposite the right angle is calledhypotenuse. It is the largest side of a right-angled triangle.

In the given triangle, the side adjacent to the angle $90-\theta$ is $y$ .

Therefore, the side adjacent to the angle $90-\theta$ is $y$ .

(c)

To determine

The reason why $\sin \theta =\cos \left(90°-\theta \right)$ .

Answer to Problem 92E

It can be said that $\sin \theta =\cos \left(90°-\theta \right)$ .

Explanation of Solution

Given Information:

The giventriangle is as follows.

  

Figure (4)

Calculation:

From the trigonometry ratios,

  $\begin{array}{c}\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}\\ \cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}\end{array}$

From the above figure.

  $\begin{array}{c}\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}\\ =\frac{y}{r}\\ \cos \left(90-\theta \right)=\frac{\text{adjacent}}{\text{hypotenuse}}\\ =\frac{y}{r}\end{array}$

From the above calculation, both the ratios are equal.

Therefore, it can be said that $\sin \theta =\cos \left(90°-\theta \right)$ .