Chapter 9.8, Problem 23E

To determine

Toexplain:The relation between cosine of one acute angle to sine of another acute angle in a right-angled triangle.

Explanation of Solution

Given information:

Right angled triangle DEF

Concept Used:

In a right-angled triangle ABC, sine for acute angle $\angle A$ is the ratio of length of side opposite to angle $\angle A$ to the length of hypotenuse and cosine of acute angle $\angle A$ is the ratio of length of side adjacent to angle $\angle A$ to the length of hypotenuse.

Calculation:

Basis definition we can say-

  $\begin{array}{l}\cos \angle D=\frac{Adjacent}{Hypotenuse}\\ \Rightarrow \cos \angle D=\frac{b}{a}\end{array}$

  $\begin{array}{l}\sin \angle F=\frac{Opposite}{Hypotenuse}\\ \Rightarrow \sin \angle F=\frac{b}{a}\end{array}$

Therefore, $\sin \angle F=\cos \angle D=\frac{b}{a}$

Hence, we can say that, in a right-angled trianglecosine of one acute angle to sine of another acute angle are equal.