Chapter 8.2, Problem 65AYU

To determine

How to solve a triangle when two sides and one angle are given?

Answer to Problem 65AYU

We will get all the information about the triangle by using law of Sine

Explanation of Solution

Given information:

What do you do first if you are asked to solve a triangle and are given two sides and the angle opposite one of them?

Calculation:

Let us consider an oblique triangle having the side $a$ , $b$ and $c$ with respect to the opposite angles $A$ , $B$ and $C$ let the two sides $a,b$ and the angle $A$ opposite to the side a is given. Now, you should follow following steps.

Calculate $\frac{\sin A}{a}\mathrm{.....}\left(1\right)$

Apply law of Sine

  $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\mathrm{.....}\left(2\right)$

In which $\frac{\sin A}{a}$ and $b$ are known.

Take first two in $\left(2\right)$ , you will get

  $\sin B=\frac{b}{a}\sin A$

  $B=\sin {\sin }^{-1}\left(\frac{b}{a}\sin A\right)or180°-{\sin }^{-1}\left(\frac{b}{a}\sin A\right)\mathrm{.......}\left(3\right)$

And verify that $A+B<180°$ here; you will get the value $\left(s\right)$ of angle $B$ .

Apply the property of the interior angles of the triangle, you will get

  $A+B+C=180°$

  $C=180°-\left(A+B\right)\mathrm{........}\left(4\right)$

Here, you will get the value $\left(s\right)$ of angle $C$ according as $B$ .

Take first and third $\left(2\right)$ , you will get

  $\frac{\sin A}{a}=\frac{\sin C}{c}$

  $c=a\times \frac{\sin C}{\sin A}\mathrm{.....}\left(5\right)$

Hence, we will get all the information about the triangle by using law of Sine in this case.