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). 3. use the formula for the area of a triangle, area = 1/2 bh, and substitute for h to get the trigonometric area formula. divide by multiply by substitute for 1/2 bc sin(a)

Explanation:

Step1: Recall sine - definition in right - triangle

In right - triangle $\triangle ABD$, $\sin(A)=\frac{h}{c}$, where $h$ is the height and $c$ is the hypotenuse.

Step2: Solve for $h$

Multiply both sides of $\sin(A)=\frac{h}{c}$ by $c$ to get $h = c\sin(A)$.

Step3: Substitute $h$ into area formula

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

Answer:

substitute for