Problem 160 - Precalculus

Question

An isosceles triangle has two congruent sides of length 9 inches. The remaining side has a length of 8 inches. Find the angle that a side of 9 inches makes with the 8-inch side.

Step-by-Step Solution

Answer

The angle is approximately 51.32 degrees.

Step 1: Understand the Triangle

In an isosceles triangle, two sides are equal in length. Here, both sides measuring 9 inches are equal, and the third side measures 8 inches. We will denote this triangle as ABC, where AB = AC = 9 inches and BC = 8 inches.

Step 2: Apply the Cosine Rule

To find the angle opposite the side BC (8 inches), we will use the cosine rule: \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \). For our triangle, this means \( 8^2 = 9^2 + 9^2 - 2 \cdot 9 \cdot 9 \cdot \cos(C) \).

Step 3: Substitute and Simplify the Equation

Substitute the values into the cosine rule equation: \[ 64 = 81 + 81 - 2 \times 9 \times 9 \times \cos(C) \]. Simplify: \[ 64 = 162 - 162 \cdot \cos(C) \].

Step 4: Solve for \( \cos(C) \)

Rearrange the equation to solve for \( \cos(C) \): \[ 162 \cdot \cos(C) = 162 - 64 \]. Therefore, \[ 162 \cdot \cos(C) = 98 \], leading to \[ \cos(C) = \frac{98}{162} \].

Step 5: Calculate \( C \) using Inverse Cosine

Calculate angle \( C \) using the inverse cosine function: \( C = \cos^{-1}\left(\frac{98}{162}\right) \). Use a calculator to find the angle.

Key Concepts

Cosine Rule

The cosine rule is an essential tool for understanding and solving triangle problems, especially when dealing with isosceles triangles. It's a generalization of the Pythagorean theorem and applies to any triangle, not just right-angled ones. The cosine rule states:

• In a triangle with sides of lengths \(a\), \(b\), and \(c\), and an angle \(C\) opposite the side \(c\), the relationship is expressed as:\[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\]

This formula is particularly useful when we know two sides of a triangle and the included angle, or when we know all three sides and are trying to find an angle.
In the given exercise, we applied the cosine rule to determine the measure of an angle in an isosceles triangle, using the sides known to us. It transforms our understanding from the less specific equal side relationship of isosceles triangles into a precise measure of an angle.

Angle Calculation

Calculating angles in a triangle, especially an isosceles one like in this exercise, can be efficiently achieved by the use of trigonometric functions. After applying the cosine rule, as discussed earlier, we found \(\cos(C)\). To get the actual angle \(C\), we use the inverse cosine function. Here's how

• After simplifying the cosine rule expression, we arrived at: \[\cos(C) = \frac{98}{162}\]
• Using a calculator, we performed the inverse operation: \[C = \cos^{-1}\left(\frac{98}{162}\right)\]
• This gives us the angle the two equal sides make with the different third side.

The inverse cosine function, often denoted as \(\cos^{-1}\), helps in retrieving the angle from the cosine value. Understanding how to use this function is key to unlocking angle measures in non-right angled triangles.

Triangle Sides

Understanding the lengths of triangle sides is crucial to solving triangle problems. An isosceles triangle, like the one in this exercise, has two sides of equal length, which affects the internal angles as well. Here is what to keep in mind:

• In our triangle, the lengths are 9 inches, 9 inches, and 8 inches.
• Knowing these lengths helps us apply the cosine rule to solve for the unknown angles.
• The unique side, often different than the equal sides, plays a critical role in angle calculations as it balances the triangle and changes the internal symmetry.

When solving triangles, always make sure each side length fits logically with the type and shape of the triangle in question. For students, it becomes a habit to visualize and verify each measure fits the properties of the triangle you are addressing.