Problem 18 - Trigonometry

Question

Each of the following problems refers to triangle \(A B C\). $$ \text { If } a=51 \mathrm{~cm}, b=24 \mathrm{~cm} \text {, and } c=41 \mathrm{~cm} \text {, find the largest angle. } $$

Step-by-Step Solution

Answer

The largest angle is approximately \(99.5^\circ\).

Step 1: Understanding the Problem

In this problem, we have triangle \(ABC\) with sides \(a = 51\) cm, \(b = 24\) cm, and \(c = 41\) cm. We need to find the largest angle in the triangle. To find the largest angle, we should start by identifying the longest side as the largest angle will be opposite the longest side.

Step 2: Identifying Longest Side

The longest side among \(a = 51\) cm, \(b = 24\) cm, and \(c = 41\) cm is \(a = 51\) cm. Hence, the largest angle will be opposite side \(a\), which is \(\angle A\).

Step 3: Using Cosine Rule

To find angle \(\angle A\), we apply the cosine rule: \[\cos A = \frac{b^2 + c^2 - a^2}{2bc}\]. Substitute the values: \(b = 24\), \(c = 41\), and \(a = 51\) into the equation.

Step 4: Substitute Values into Formula

Substitute the values into the formula: \[\cos A = \frac{24^2 + 41^2 - 51^2}{2 \times 24 \times 41}\]. Calculate the values of \(24^2\), \(41^2\), and \(51^2\).

Step 5: Calculating Values

Calculate the values: \(24^2 = 576\), \(41^2 = 1681\), and \(51^2 = 2601\). Then perform the substitution: \[\cos A = \frac{576 + 1681 - 2601}{1968}\].

Step 6: Simplifying Equation

Simplify the equation: \[\cos A = \frac{-344}{1968}\]. Divide both the numerator and the denominator by 8 to simplify: \[\cos A = \frac{-43}{246}\].

Step 7: Finding the Angle

Find \(\angle A\) by taking the arccosine of the result: \(\angle A = \cos^{-1}\left(\frac{-43}{246}\right)\). Calculate this using a calculator to find \(\angle A\).

Step 8: Result Interpretation

Calculate \(\angle A \approx 99.5^\circ\). Hence, \(\angle A\) is the largest angle in triangle \(ABC\).

Key Concepts

Understanding Triangle Sides

In any triangle, the three sides are key components that determine its shape, size, and the measures of its angles. The sides of the triangle are usually denoted as lowercase letters - often corresponding to the vertex opposite that side. For instance, in triangle \(ABC\), the sides are labeled as \(a\), \(b\), and \(c\). Here, they represent the lengths opposite vertices \(A\), \(B\), and \(C\), respectively.
A crucial principle in triangles is that the side opposite the largest angle is the longest, and conversely, the side opposite the smallest angle is the shortest. This is the reason why identifying the largest side is an effective step in finding the largest angle.
From the dimensions given in the problem, side \(a = 51\) cm is clearly the greatest of the three sides. Armed with this knowledge, we can ascertain that angle \(A\), opposite to this side, will be the largest angle in the triangle.

Identifying the Largest Angle

To determine the largest angle in a triangle, we first focus on identifying the longest side, as it is directly opposite the largest angle. This is guided by a fundamental property of triangles that aligns sides with their opposite angles: the bigger the side, the larger the opposite angle gets.
Given sides \(a = 51\) cm, \(b = 24\) cm, and \(c = 41\) cm, the largest angle \(\angle A\) corresponds to the largest side \(a = 51\) cm. To find this angle with accuracy, the Law of Cosines becomes a necessary tool. The Law of Cosines provides us with a formula that relates the lengths of triangle sides with one of its interior angles.

• The formula is: \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\).
• This formula is a powerful resource as it allows us to calculate the angle, even without knowing the other angles right away.

This strategic approach simplifies finding an angle by connecting it directly to the side lengths we already know.

Trigonometry and Problem-Solving

Trigonometry is a branch of mathematics focusing on the specific relationships between the sides and angles of triangles. In scenarios like ours, where the goal is to solve for an angle in a non-right triangle, trigonometric rules like the Law of Cosines play a vital role.
The Law of Cosines is especially useful when dealing with non-right triangles and when you know the lengths of all three sides. It offers a straightforward way to compute an angle, which is instrumental in further exploring the triangle’s properties.
Here's a breakdown of how it's utilized in our problem:

• Start by substituting known values into the Law of Cosines: \(b = 24\), \(c = 41\), and \(a = 51\).
• Calculate \(b^2 = 576\), \(c^2 = 1681\), and \(a^2 = 2601\), and use these in the formula to find \(\cos A\).
• Simplify the expression to find \(\cos A = \frac{-344}{1968}\).
• Further simplify to get \(\cos A = \frac{-43}{246}\).

Finally, apply the inverse cosine function to extract the angle, showing that \(\angle A \approx 99.5^\circ\). This exemplifies how trigonometry can precisely solve problems involving the geometry of triangles.