Question

follow these steps to derive the trigonometric area formula.

1. for △abd, sin(a) = h/c
2. solve the previous equation for h

h = c sin(a)

1. use the formula for the area of a triangle, area = 1/2 bh, and substitute h to get the trigonometric area formula

area = 1/2 bc sin(a)

Explanation:

Step1: Recall sine - definition

In right - triangle $\triangle ABD$, $\sin(A)=\frac{h}{c}$, so $h = c\sin(A)$.

Step2: Substitute $h$ into area formula

The area of a triangle is $A=\frac{1}{2}bh$. Substitute $h = c\sin(A)$ into it. We get $A=\frac{1}{2}b\times c\sin(A)=\frac{1}{2}bc\sin(A)$.

Answer:

$\frac{1}{2}bc\sin(A)$