Problem 43 - Trigonometry

Question

Find the area of triangle \(A B C\) if \(A=56^{\circ}, B=71^{\circ}\), and \(c=21\) inches. a. \(150 \mathrm{in}^{2}\) b. \(200 \mathrm{in}^{2}\) c. 220 in \(^{2}\) d. \(240 \mathrm{in}^{2}\)

Step-by-Step Solution

Answer

The area of triangle \(ABC\) is approximately 220 \(\mathrm{in}^2\).

Step 1: Check Angle Sum in a Triangle

To find the missing angle of the triangle, use the fact that the sum of angles in a triangle is 180 degrees. Since \(A = 56^{\circ}\) and \(B = 71^{\circ}\), we can find \(C\) as follows: \(C = 180^{\circ} - 56^{\circ} - 71^{\circ}\).

Step 2: Calculate Angle C

Perform the calculation: \(C = 180^{\circ} - 127^{\circ} = 53^{\circ}\). Thus, the measure of angle \(C\) is \(53^{\circ}\).

Step 3: Use the Law of Sines to Find Side Lengths

According to the Law of Sines, \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). Since we know \(c = 21\) inches, \(A = 56^{\circ}\), \(B = 71^{\circ}\), and \(C = 53^{\circ}\), we use this to find \(a\) and \(b\) if needed, but primarily to find the area.

Step 4: Calculate Area Using the Known Elements

We can use the formula for the area of a triangle given two angles and the included side: \[ \text{Area} = \frac{1}{2} \cdot c^2 \cdot \frac{\sin A \cdot \sin B}{\sin C} \]Substituting, we have: \[ \text{Area} = \frac{1}{2} \cdot 21^2 \cdot \frac{\sin 56^{\circ} \cdot \sin 71^{\circ}}{\sin 53^{\circ}} \]

Step 5: Perform Calculation for Area

Calculate the sine values using a calculator:- \(\sin 56^{\circ} \approx 0.8290\) - \(\sin 71^{\circ} \approx 0.9455\)- \(\sin 53^{\circ} \approx 0.7986\)Substituting these into the formula:\[ \text{Area} = \frac{1}{2} \cdot 441 \cdot \frac{0.8290 \cdot 0.9455}{0.7986} \]

Step 6: Final Area Calculation

Perform the multiplication inside the fraction first:\(0.8290 \times 0.9455 = 0.7835355\)Then divide by \(0.7986\): \(\frac{0.7835355}{0.7986} \approx 0.9811\)Finally, calculate the area: \[ \text{Area} = \frac{1}{2} \cdot 441 \cdot 0.9811 \approx 216.22005 \]This rounds to 216 \(\mathrm{in}^2\), but the closest option to this value is 220 \(\mathrm{in}^2\).

Key Concepts

Law of Sines

The Law of Sines is an essential trigonometric principle used in solving triangle problems. It relates the lengths of a triangle's sides to the sines of its opposite angles. Here's the basic formula:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

In triangle problems, like finding areas or unknown sides, this law provides a handy way to relate known and unknown quantities. For example, if you have a triangle with a known angle and its opposite side, along with another angle, you can find the third side using this law. It's especially helpful when dealing with non-right triangles.
In our exercise, we utilized the Law of Sines to connect the known side, \( c = 21 \) inches, and the given angles \( A = 56^{\circ} \), \( B = 71^{\circ} \), and the calculated \( C = 53^{\circ} \) to find the area effectively.

Angle Sum in a Triangle

The angle sum property of a triangle is straightforward but vital. In any triangle, the sum of the interior angles equals 180 degrees. This basic rule is pivotal when you know two angles and need to find the third.

• Formally, for a triangle with angles \( A \), \( B \), and \( C \):
• \( A + B + C = 180^{\circ} \)

In practical terms, if a student knows two angles, they can always calculate the third by subtracting the sum of the known angles from 180 degrees. In the provided problem, with angles \( A = 56^{\circ} \) and \( B = 71^{\circ} \), calculating \( C \) was our first step by subtracting the sum of \( A \) and \( B \) from 180 degrees, giving us \( C = 53^{\circ} \).
This foundational property is critical to ensure we have all necessary angles for further calculations, such as using the Law of Sines.

Trigonometric Ratios

Trigonometric ratios are relationships between the angles and sides of triangles, used extensively in triangle calculations. In particular, when dealing with non-right triangles, these ratios help compute unknown sides or angles. The primary trigonometric functions are sine, cosine, and tangent. Each relates a side of a triangle to its angle as follows:

• Sine (sin): Opposite side over hypotenuse
• Cosine (cos): Adjacent side over hypotenuse
• Tangent (tan): Opposite side over adjacent side

In our area calculation, we used the sine function specifically, since it directly relates the angle to the side opposite and is used extensively in the Law of Sines. Accurate computation of sine values was essential for our area formula:

• \( \text{Area} = \frac{1}{2} \cdot c^2 \cdot \frac{\sin A \cdot \sin B}{\sin C} \)

Knowing these ratios and how to apply them is crucial for solving many trigonometric problems involving triangles.