Chapter 8, Problem 8CUR

To determine

To find : the value of cos B

Answer to Problem 8CUR

The value of cos B is $\cos B=\frac{15}{17}$

Explanation of Solution

Given Information: A right angled triangle with angle B such that, $\sin B=\frac{8}{17}$

Formula used:

1) In a right triangle the sine of the any angle is the ratio of the measure of side opposite to the given angle to the hypotenuse of triangle. The cosine of the any angle is the ratio of the adjacent side of the given angle to the hypotenuse of triangle.

  

It can be expressed as,

  $\begin{array}{l}\sin B=\frac{a}{c}\\ \cos B=\frac{b}{c}\end{array}$

2) The relationship between the legs of right triangle (shown in figure below) a , b and its hypotenuse c is obtained by using Pythagoras theorem.

It can be expressed as,

  $\begin{array}{l}{c}^2=a^2+b^2\\ c=\sqrt{a^2+b^2}\end{array}$

Calculation: Consider the right triangle given below.

  

Here, $\sin B=\frac{8}{17}$ . So, with reference to the definition of sine of an angle, this implies, the opposite side to the angle B is 8 and hypotenuse is 17 as shown in the above Figure.

Now, cosine of the angle is defined as,

  $\cos B=\frac{x}{17}$

So, the length of the adjacent side x can be calculated using the Pythagoras theorem as,

  $\begin{array}{c}x=\sqrt{{17}^2-8^2}\\ =\sqrt{225}\\ =15\end{array}$

Therefore, cosine of B is given by,

  $\cos B=\frac{15}{17}$