Derive a formula for the area of the triangle, given b, A, B, and C. That is, derive the formula: (b^2sinAsinC)/(2sinB)

To derive the formula for the area of a triangle given the base $b$ and angles $A$, $B$, and $C$, we can utilize the sine rule and the definition of the area of a triangle. Let's break this down step by step.

Understanding Triangle Area

The area $A$ of a triangle can generally be calculated using the formula:

Area = 0.5 × base × height

However, when we have angles and one side, we can use a different approach involving the sine of the angles.

Using the Sine Rule

The sine rule states that for any triangle with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, the following relationship holds:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

From this, we can express the sides in terms of the angles and the known side $b$:

$a=\frac{b\cdot \sin A}{\sin B}$

$c=\frac{b\cdot \sin C}{\sin B}$

Finding the Area

Now, we can use the formula for the area of a triangle in terms of two sides and the included angle. The area can also be expressed as:

Area = 0.5 × a × c × \sin B

Substituting the expressions for $a$ and $c$ from the sine rule, we get:

Area = 0.5 × \left(\frac{b \cdot \sin A}{\sin B}\right) × \left(\frac{b \cdot \sin C}{\sin B}\right) × \sin B

Simplifying the Expression

Now, let's simplify this expression:

• First, notice that one $\sin B$ in the denominator cancels with $\sin B$ in the numerator:

Area = 0.5 × \frac{b^2 \cdot \sin A \cdot \sin C}{\sin B}

Multiplying both sides by 2 gives us:

Area = \frac{b^2 \cdot \sin A \cdot \sin C}{2 \cdot \sin B}

Final Formula

Thus, the derived formula for the area of the triangle, given the base $b$ and angles $A$, $B$, and $C$, is:

Area = \frac{b^2 \cdot \sin A \cdot \sin C}{2 \cdot \sin B}

Practical Example

For instance, if you have a triangle where $b=10$, $A={30}^{\circ }$, $B={60}^{\circ }$, and $C={90}^{\circ }$, you can substitute these values into the formula:

• Calculate $\sin A=\sin {30}^{\circ }=0.5$
• Calculate $\sin C=\sin {90}^{\circ }=1$
• Calculate $\sin B=\sin {60}^{\circ }\approx 0.866$

Plugging these into the formula gives:

Area = \frac{10^2 \cdot 0.5 \cdot 1}{2 \cdot 0.866} \approx 28.87

This process illustrates how you can derive the area of a triangle using angles and one side, providing a powerful tool for solving various problems in geometry.