Problem 32 - Precalculus

Question

Find the area of each triangle with measures given. $$b=2 \sqrt{2}, c=4, \beta=45^{\circ}$$

Step-by-Step Solution

Answer

The area of the triangle is 4 square units.

Step 1: Use the Area Formula for Triangles

The area of a triangle can be calculated with the formula: \[ \text{Area} = \frac{1}{2}bc \sin(\beta) \]where \(b\) and \(c\) are the sides, and \(\beta\) is the angle between them.

Step 2: Substitute the Given Values

Substitute the given values into the formula: \[ \text{Area} = \frac{1}{2} \times 2 \sqrt{2} \times 4 \times \sin(45^{\circ}) \]

Step 3: Calculate \( \sin(45^{\circ}) \)

The sine of \(45^{\circ}\) is \(\sin(45^{\circ}) = \frac{\sqrt{2}}{2}\). Substitute this value into the formula.

Step 4: Simplify the Calculation

Replace \(\sin(45^{\circ})\) in the area formula:\[ \text{Area} = \frac{1}{2} \times 2 \sqrt{2} \times 4 \times \frac{\sqrt{2}}{2} \]Now, simplify the expression:First, \(2 \sqrt{2} \times \frac{\sqrt{2}}{2} = 2 \frac{{\sqrt{2} \times \sqrt{2}}}{2} = 2 \times \frac{2}{2} = 2\).Then, \( 4 \times 2 = 8\).Finally, multiply by \(\frac{1}{2}\):\[ \frac{1}{2} \times 8 = 4 \]

Step 5: Conclude with the Area Result

Thus, the area of the triangle is 4 square units.

Key Concepts

Area Formula for Triangles

When calculating the area of triangles, one of the most efficient methods involves using two sides and the sine of the included angle. This is particularly helpful when you do not have access to the base or height. The formula used is:

• \( \text{Area} = \frac{1}{2}bc \sin(\beta) \)

Here, \(b\) and \(c\) are the lengths of two sides, while \(\beta\) is the angle between them. This formula leverages trigonometric properties to provide an accurate measure of the triangle's area, making it extremely practical in geometry and various applications involving triangular assessment.
This formula offers a direct computation without requiring perpendicular measurements, simplifying calculations considerably in many triangle-related problems.

Sine Function in Geometry

The sine function is essential in geometry when dealing with non-right angled triangles. In the context of calculating the area, the sine of an angle provides a ratio that corresponds to the opposite side and the hypotenuse of a right triangle. In practical terms, it's often used to discover unknown elements of a triangle.
For example, knowing \(\beta = 45^{\circ}\), we apply the sine function:

• \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \)

This value is crucial for determining areas and sides in cases where direct measurements aren't available. Understanding this function allows those studying geometry to approach problems from multiple angles, literally and figuratively.

Trigonometric Functions in Calculations

Trigonometric functions, including sine, cosine, and tangent, form a bedrock in solving many mathematical problems, especially in triangle calculations. They provide the means to connect angles and side lengths in triangles, allowing us to find unknown measurements easily.
In the scope of area calculation, the function \(\sin\) is particularly pivotal. The reason being, it helps transform the angle's geometric significance into a calculable unit:

• The relation \( \text{Area} = \frac{1}{2}bc \sin(\beta) \) allows the evaluation of triangles where direct side measurements don't establish the base-height configuration.

By understanding trigonometric functions, students enhance their ability to solve problems not just in theoretical exercises but in real-world situations where such geometric calculations are necessary.