Question

utilizing the trigonometric triangle area formula
in △pqr, the obtuse angle is 105°, and the shorter sides measure 4 and 7 units.
rounded to the nearest tenth, what is the area of △pqr?
area = square units
13.5
14.0
27.0
28.0

Explanation:

Step1: Recall the formula

The area formula for a triangle using trigonometry is $A=\frac{1}{2}ab\sin C$, where $a$ and $b$ are two - side lengths of the triangle and $C$ is the included angle.

Step2: Identify the values

Here, $a = 4$, $b = 7$, and $C=105^{\circ}$. First, find $\sin(105^{\circ})=\sin(60^{\circ}+45^{\circ})=\sin60^{\circ}\cos45^{\circ}+\cos60^{\circ}\sin45^{\circ}=\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}+\frac{1}{2}\times\frac{\sqrt{2}}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.966$.

Step3: Calculate the area

Substitute the values into the formula: $A=\frac{1}{2}\times4\times7\times\sin(105^{\circ})$.
$A = 14\times\sin(105^{\circ})$.
$A\approx14\times0.966 = 13.524\approx13.5$.

Answer:

13.5 square units