Question

trigonometric area formula: \\(\text{area} = \frac{1}{2}ab\sin(c)\\)

what is the area of triangle pqr? round to the nearest tenth of a square unit.

\\(\bigcirc\\) 70.5 square units
\\(\bigcirc\\) 111.3 square units
\\(\bigcirc\\) 185.4 square units
\\(\bigcirc\\) 222.5 square units

Explanation:

🆕 New Concept Discovered: Trigonometric Area of a Triangle
Using two sides and the angle between them to find area.

Step 1: Identify the given values

To use the trigonometric area formula, we need two side lengths and the measure of the included angle (the angle formed between those two sides).

From the given triangle \(PQR\):

• Side \(PQ = 20\)
• Side \(PR = 12\)
• The angle between these two sides is \(\angle P = 68^\circ\)

Step 2: Apply the area formula

The formula for the area of a triangle using trigonometry is:

$$ \text{Area} = \frac{1}{2} a b \sin(C) $$

Substitute our known values into the formula:

$$ \text{Area} = \frac{1}{2} \times 20 \times 12 \times \sin(68^\circ) $$

Step 3: Calculate the final value

Simplify the multiplication:

$$ \text{Area} = 10 \times 12 \times \sin(68^\circ) $$

$$ \text{Area} = 120 \times \sin(68^\circ) $$

Using a calculator to find \(\sin(68^\circ) \approx 0.92718\):

$$ \text{Area} \approx 120 \times 0.92718 $$

$$ \text{Area} \approx 111.26 $$

Rounding to the nearest tenth of a square unit gives \(111.3\).

Answer:

111.3 square units