Question

State the formulas for the area of a triangle ABC that were derived using the law of sines.

Answer 1

Given

a triangle ABC

Find

Area of a triangle using the laws of sines

Explanation

As area of triangle =

$\frac{1}{2}\times base\times height$

since, it is a right angle triangle.

Assume the sine of angle A

$sinA=\frac{opposite\text{ side}}{hypotenuse}$

so, we have

$sinA=\frac{h}{c}$

hence,

$h=csinA$

now, put the value of h in area of triangle,

$Area\text{ of triangle =}\frac{1}{2}\times b\times csinA=\frac{1}{2}bcsinA$

similarly we can write for the sinB and SinC

$\frac{1}{2}absinC\text{ and }\frac{1}{2}acsinB$

Final Answer

Area of triangle in the law of sines is

$\frac{1}{2}bcsinA$