Problem 77 - Algebra and Trigonometry

Question

Problems \(77-86\) are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the area of the triangle with \(a=8, b=11\), and \(C=113^{\circ}\).

Step-by-Step Solution

Answer

The area is approximately 40.788 square units.

Step 1 - Understand the Problem

The problem provides the lengths of two sides of a triangle, denoted as a and b, and the measure of the included angle, denoted as C. The aim is to find the area of the triangle.

Step 2 - Recall the Formula for the Area of a Triangle

The area of a triangle when two sides and the included angle are known can be found using the formula: \[ \text{Area} = \frac{1}{2} \times a \times b \times \text{sin}(C) \]

Step 3 - Substitute the Given Values

Substitute the provided values into the formula: \[ a = 8, \, b = 11, \, C = 113^{\text{circ}} \] This leads to: \[ \text{Area} = \frac{1}{2} \times 8 \times 11 \times \text{sin}(113^{\text{circ}}) \]

Step 4 - Calculate the Sine of the Angle

Use a calculator to find the sine of 113 degrees: \[ \text{sin}(113^{\text{circ}}) \rightarrow \text{sin}(113^{\text{circ}}) \thickapprox 0.927 \]

Step 5 - Compute the Area

Plug the sine value into the area formula: \[ \text{Area} = \frac{1}{2} \times 8 \times 11 \times 0.927 \] Simplify the multiplication step-by-step: \[ \text{Area} = 4 \times 11 \times 0.927 = 44 \times 0.927 = 40.788 \]

Key Concepts

triangle area formula

Understanding how to calculate the area of a triangle when you know two sides and the included angle is essential in trigonometry. One effective formula for this purpose is:

\[ \text{Area} = \frac{1}{2} \times a \times b \times \text{sin}(C) \]

Here:

• \(a\) and \(b\) are the lengths of two sides.
• \(C\) is the measure of the included angle between these sides.

This formula leverages the sine function, enabling us to find the triangle's area even without knowing the height.

To employ this formula properly:

• Ensure the angle measurement is in degrees (or convert it if it's in radians).
• Use a calculator for complex sine computations, particularly for non-standard angles.

sine function

The sine function is a critical tool in trigonometry, especially for solving triangle-related problems. It relates to a right-angled triangle, where it measures the ratio of the opposite side to the hypotenuse for a given angle.

For an angle \( \theta \): \[ \text{sin}(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]

In non-right-angled triangles, such as in our problem, we use the sine function along with known sides and an included angle to find other properties, like area.

Key points when using the sine function:

• The angle must be in degrees unless specified otherwise.
• Advanced calculators or trigonometric tables are useful for obtaining sine values of any angle.

For the given problem, the sine value for \(113^{\circ}\) is approximately \(0.927\). This small but crucial step turns our formula into a numerical value that simplifies further calculation.

included angle

The concept of the included angle is fundamental in trigonometry problems involving triangles. This is the angle formed between two known side lengths.

To visualize: If we label a triangle with sides \(a\) and \(b\), and the angle between them is \(C\), then \(C\) is the included angle.

This angle is essential because it directly influences the calculation of the triangle's area. By knowing the included angle, we convert its sine value into a critical multiplier in the area formula: \[ \text{Area} = \frac{1}{2} \times a \times b \times \text{sin}(C) \]

This means that both the magnitude of the sides and the measure of the included angle determine the size of the triangle's area.

When working on problems like this, always ensure the angle provided directly sits between the two known sides. Misidentifying which angle is included can lead to incorrect solutions and confuse your results.