Problem 8 - Algebra 2 and Trigonometry

Question

In \(7-12,\) find the cosine of each angle of the given triangle. In \(\triangle A B C, a=12, b=8, c=8\)

Step-by-Step Solution

Answer

\(\cos A = -\frac{1}{8}, \cos B = \frac{3}{4}, \cos C = \frac{3}{4}\).

Step 1: Identify Triangle Type

Since two sides of the triangle are equal \(b = c\), this is an isosceles triangle. We can use this information to simplify our calculations.

Step 2: Use the Cosine Rule on Angle A

To find \cos A\, use the cosine rule: \cos A = \frac{b^2 + c^2 - a^2}{2bc}\. Substitute the values \(b = 8\), \(c = 8\), and \(a = 12\) into the equation: \\[\cos A = \frac{8^2 + 8^2 - 12^2}{2 \times 8 \times 8} = \frac{64 + 64 - 144}{128} = \frac{-16}{128} = -\frac{1}{8}.\]

Step 3: Use the Cosine Rule on Angles B and C

Since the triangle is isosceles \(b = c\), angles \(B\) and \(C\) are equal. Use the cosine rule for angle \(B\): \cos B = \frac{c^2 + a^2 - b^2}{2ca}\. Substitute the values: \\[\cos B = \frac{8^2 + 12^2 - 8^2}{2 \times 8 \times 12} = \frac{64 + 144 - 64}{192} = \frac{144}{192} = \frac{3}{4}.\]

Step 4: Conclude with the Cosine of Angle C

Similarly, since angles \(B\) and \(C\) are equal, \cos C = \cos B = \frac{3}{4}\.

Key Concepts

Isosceles Triangle

An isosceles triangle is a type of triangle that has at least two sides of equal length. In triangle ABC, sides b and c are both 8, indicating that it is an isosceles triangle. This unique property brings along certain symmetrical qualities:

• Two angles opposite the equal sides are also equal.
• It simplifies calculations in trigonometry because once you know one angle, the corresponding angle can be easily determined.

In our example, angles B and C are equal since their opposite sides b and c are the same in length. These properties make isosceles triangles a fundamental part of geometry and particularly useful in problems involving symmetry.

Cosine of an Angle

The cosine of an angle in a triangle is related to the lengths of the sides of the triangle. Cosine is one of the primary trigonometric ratios, defined as the adjacent side over the hypotenuse in a right-angled triangle. However, for non-right-angled triangles, such as our isosceles triangle, we use the Cosine Rule:

• The rule helps to determine the cosine of an angle when we know the lengths of all three sides.
• The formula is: \( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \)

This formula perfectly serves our problem, where we need to find the cosine of angles in a non-right triangle. By plugging the side lengths into the formula, we retrieved the values of \( \cos A \) and \( \cos B \) without needing any angle measuring tools.

Triangle Types

Triangles are classified by either their side lengths or their angle measures. Understanding the types of triangles is crucial in solving geometric problems effectively.

• Equilateral Triangle: All sides and angles are equal.
• Isosceles Triangle: As previously detailed, two sides are equal, and the angles opposite these sides are also equal.
• Scalene Triangle: All sides and angles are different.
• Right Triangle: Has one 90-degree angle.

Each type of triangle offers specific properties that can simplify calculation and play a crucial role in solving triangle problems, especially in trigonometry. Recognizing an isosceles triangle in our task aided in reducing complexity when calculating angles using the cosine rule.