Problem 36 - Precalculus

Question

Find the area of each triangle with measures given. $$a=6.3, b=4.8, \gamma=17^{\circ}$$

Step-by-Step Solution

Answer

The area of the triangle is approximately 4.42 square units.

Step 1: Understand the Problem

We need to find the area of a triangle given two sides and the included angle. We'll use the formula for the area of a triangle when two sides and an included angle are known: \[ A = \frac{1}{2}ab\sin(\gamma) \] where \(a\) and \(b\) are sides of the triangle and \(\gamma\) is the included angle.

Step 2: Substitute the Known Values

From the problem, we have \( a = 6.3 \), \( b = 4.8 \), and \( \gamma = 17^{\circ} \). Substitute these values into the area formula:\[ A = \frac{1}{2} \times 6.3 \times 4.8 \times \sin(17^{\circ}) \]

Step 3: Calculate \( \sin(17^{\circ}) \)

Find the sine of \(17^{\circ}\). Using a calculator, \( \sin(17^{\circ}) \approx 0.2924 \).

Step 4: Compute the Product

Now substitute \( \sin(17^{\circ}) \approx 0.2924 \) into the equation:\[ A = \frac{1}{2} \times 6.3 \times 4.8 \times 0.2924 \] Calculate \( 6.3 \times 4.8 \times 0.2924 \approx 8.8387 \).

Step 5: Calculate the Area

Finally, calculate the area using the expression from the last step:\[ A = \frac{1}{2} \times 8.8387 \approx 4.41935 \] Round the answer to an appropriate number of decimal places: \( A \approx 4.42 \).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It's an essential tool when it comes to solving problems involving triangles, especially when working with non-right triangles. By understanding trigonometry, we can calculate unknown lengths and angles using trigonometric ratios such as sine, cosine, and tangent.

In the context of this exercise, trigonometry helps us to determine the area of non-right triangles. When two sides and an included angle are known, traditional methods may not suffice. Instead, we use trigonometry to find the area using the formula:

• \[A = \frac{1}{2}ab\sin(\gamma)\]

where \(a\) and \(b\) are the lengths of two sides, and \(\gamma\) is the included angle. This formula showcases how trigonometry seamlessly blends algebra and geometry to solve complex problems.

Sine Function

The sine function is a foundational trigonometric function that relates a right triangle's angle to the ratio of the length of the opposite side to the hypotenuse. In simpler terms, it provides a way to measure how much a specific angle opens up.In triangles, especially those that aren't right-angled, the sine function becomes crucial in measuring areas. In our exercise, the sine of the angle \(\gamma = 17^{\circ}\) is calculated to find the triangle's area. The sine function is extracted from trigonometry tables or calculators and is fundamental because it helps compute:

• The vertical height related to the base length of a triangle
• The effective use of angles in geometric computations
• Revenue within the formula \[A = \frac{1}{2}ab\sin(\gamma)\]

In this exercise, we found \(\sin(17^{\circ}) \approx 0.2924\), which helps to calculate area along with given side lengths, showing the sine function's vital role in non-right triangle scenarios.

Geometry

Geometry is the branch of mathematics dealing with shapes, sizes, and the properties of space. When solving problems related to triangles, understanding geometry is paramount to evaluating different properties of shapes.

In terms of this exercise, geometry explains why we use specific formulas for determining areas. The concept of area is essential in geometry, denoting the surface a shape covers. For triangles, there are various formulas depending on the information available:

• The basic formula for area in right triangles: \( \frac{1}{2} \times \text{base} \times \text{height} \)
• The use of trigonometry for non-right triangles: \(\frac{1}{2}ab\sin(\gamma)\)
• The Heron's formula for when all sides are known

By integrating geometry, trigonometry, and the sine function, we extract meaningful solutions for various shapes, extending our problem-solving capabilities. The concept of area underscores geometry's role in real-world applications and theoretical frameworks.