Problem 34 - Precalculus

Question

Find the area of each triangle with measures given. $$b=9, c=10, \alpha=100^{\circ}$$

Step-by-Step Solution

Answer

The area of the triangle is approximately 44.316 square units.

Step 1: Understanding the Problem

To find the area of a triangle when given two sides and the included angle, we'll use the formula: \[ A = \frac{1}{2}bc \sin(\alpha) \] where \(b\) and \(c\) are the sides and \(\alpha\) is the included angle between those sides.

Step 2: Convert the Angle to Radians

The trigonometric functions in most calculators use angles in radians. Therefore, convert the angle \(\alpha = 100^{\circ}\) to radians: \[ \alpha = \frac{100 \times \pi}{180} = \frac{5\pi}{9} \text{ radians} \].

Step 3: Calculate \(\sin(\alpha)\)

Compute the sine of the converted angle: \[ \sin(\frac{5\pi}{9}) \]. Use a calculator to find its approximate value: \(\sin(\frac{5\pi}{9}) \approx 0.9848\).

Step 4: Substitute Values into the Formula

Plug the known values into the area formula: \[ A = \frac{1}{2} \times 9 \times 10 \times 0.9848 \]. Simplify to solve: \[ A \approx 44.316 \text{ square units} \].

Step 5: Finalize the Answer

The final computed area value is an approximation due to rounding of \(\sin(\alpha)\). Therefore, the approximate area of the triangle is 44.316 square units.

Key Concepts

Trigonometric Functions

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. In this context, trigonometric functions like sine, cosine, and tangent help us solve various geometric problems. For calculating the area of a triangle, when two sides and the included angle are given, the sine function is particularly useful. This is because the area can be expressed with the formula \( A = \frac{1}{2}bc \sin(\alpha) \) where \( b \) and \( c \) are the sides, and \( \alpha \) is the angle between them.

• The sine function, \( \sin(\alpha) \), takes an angle \( \alpha \) and returns a value between -1 and 1.
• This value represents the relationship between the opposite side and the hypotenuse in a right triangle, but it's extended to any angle in terms of the unit circle.

Understanding how to use trigonometric functions effectively requires practice. With the help of calculators, especially those capable of calculating sine values in radians or degrees, trigonometry becomes a powerful tool for solving real-world geometry problems.

Radians Conversion

When working with trigonometric functions, it's crucial to understand how angle measurements work in radians. Many calculators and mathematical formulas use radians rather than degrees. One full circle equals \( 2\pi \) radians, which is equivalent to 360 degrees. Thus, converting degrees to radians often simplifies calculations.

• To convert degrees to radians, use the conversion factor \( \pi/180 \).
• For example, to convert \( 100^{\circ} \) to radians, you calculate \( 100 \times \pi/180 = \frac{5\pi}{9} \) radians.

This conversion is essential in trigonometry because it aligns angle measurements with the standard inputs for many scientific calculators, enabling accurate computations of trigonometric functions.

Triangle Sides and Angles

Triangles are geometric figures with three sides and three angles. When calculating areas in triangles where direct altitude and base calculations aren't straightforward, the included angle method becomes very handy.

• The sides of the triangle are denoted as \( b \) and \( c \) in this context.
• The angle formed between these sides is \( \alpha \).

The formula \( A = \frac{1}{2}bc \sin(\alpha) \) gives us the area by focusing on the two given sides and the included angle—an elegant solution without needing to know the height directly. The included angle affects the resulting value directly by scaling the areas formed from vectors \( b \) and \( c \) in the plane of the triangle. Understanding these basics helps one tackle more complex problems involving other types of triangles and their properties.