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Chapter 5: Problem 98

Find the area of the triangle with \(a=3.6, b=4.5,\) and \(\gamma=37.1^{\circ}\).

Short Answer

Expert verified
The area of the triangle is approximately 4.68 square units.

Step by step solution

01

- Understand the Given Data

Identify the given sides and angle of the triangle: side lengths are \(a = 3.6\) and \(b = 4.5\), and the included angle \(\gamma = 37.1^{\circ}\).
02

- Recall the Area Formula

Use the formula for the area of a triangle when two sides and the included angle are known: \[ \text{Area} = \frac{1}{2} \times a \times b \times \text{sin}(\theta) \]. Here, \(a\) and \(b\) are the side lengths, and \(\theta\) is the included angle.
03

- Substitute the Given Values

Substitute the given values into the formula: \[ \text{Area} = \frac{1}{2} \times 3.6 \times 4.5 \times \text{sin}(37.1^{\circ}) \].
04

- Calculate sin(37.1掳)

Find the sine of the angle: \( \text{sin}(37.1^{\circ}) \approx 0.602 \).
05

- Perform the Multiplication

Multiply the values together: \[ \text{Area} = \frac{1}{2} \times 3.6 \times 4.5 \times 0.602 \].
06

- Final Calculation

Complete the calculation: \[ \text{Area} = \frac{1}{2} \times 3.6 \times 4.5 \times 0.602 \ \text{Area} = \frac{1}{2} \times 9.72 \times 0.602 \ \text{Area} = 4.67892 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

triangle area formula
To find the area of a triangle when two sides and the included angle are known, we use a specific formula: \[ \text{Area} = \frac{1}{2} \times a \times b \times \text{sin}(\theta) \] This formula is very useful because it allows us to determine the area without knowing the height. Here: \[ a \text{ and } b \text{ are the side lengths, and } \theta \text{ is the included angle} \] Using this method, we can quickly find the area as long as we have the correct values for the sides and the angle.
sine function in trigonometry
The sine function is a fundamental concept in trigonometry. It relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. For an angle \(\theta\), the sine function is written as \(\text{sin}(\theta)\).
In our given problem, we have \(\gamma = 37.1^{\circ}\). So, we need to find \(\text{sin}(37.1^{\circ})\).
Using a calculator, we find that: \[ \text{sin}(37.1^{\circ}) \approx 0.602 \] This value is then used in our area formula to find the area of the triangle.
angles in triangles
Understanding the angles in triangles is key to solving many geometric problems. In any triangle, the sum of the three interior angles is always 180 degrees. For our exercise, we were given one angle, \(\gamma = 37.1^{\circ}\), which is known as the included angle because it is between the two known sides. By knowing this angle and the sides, we can utilize the sine function to calculate the area.
It's important to get familiar with how different angles relate to each other in various types of triangles, including right, acute, and obtuse triangles.

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