Problem 14 - Trigonometry

Question

Each of the following problems refers to triangle \(A B C\). $$ \text { If } a=22 \mathrm{yd}, b=24 \mathrm{yd} \text {, and } c=29 \mathrm{yd} \text {, find the largest angle. } $$

Step-by-Step Solution

Answer

The largest angle is approximately \(77.80^{\circ}\).

Step 1: Identify the largest side

In a triangle, the largest angle is opposite the largest side. Compare the given side lengths: \(a = 22\, \text{yd}, b = 24\, \text{yd}\), and \(c = 29\, \text{yd}\). The largest side is \(c\).

Step 2: Apply the Law of Cosines

We will use the Law of Cosines to find the angle opposite the largest side \(c\). The formula is: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] Substitute the values: \[ 29^2 = 22^2 + 24^2 - 2 \cdot 22 \cdot 24 \cdot \cos(C) \]

Step 3: Calculate the squares of the sides

Compute the squares: \[ 29^2 = 841, \quad 22^2 = 484, \quad 24^2 = 576 \] Substituting these into the equation gives: \[ 841 = 484 + 576 - 2 \cdot 22 \cdot 24 \cdot \cos(C) \]

Step 4: Simplify the equation

Add the squares of sides \(a\) and \(b\): \[ 841 = 1060 - 2 \cdot 22 \cdot 24 \cdot \cos(C) \]

Step 5: Solve for \(\cos(C)\)

Rearrange to isolate \(\cos(C)\): \[ 841 - 1060 = -2 \cdot 22 \cdot 24 \cdot \cos(C) \] \[ -219 = -2 \cdot 528 \cdot \cos(C) \] Divide by \(-1056\) to solve for \(\cos(C)\): \[ \cos(C) = \frac{219}{1056} \approx 0.2074 \]

Step 6: Determine the largest angle

Finally, find the angle \(C\) using \(\cos^{-1}\): \[ C = \cos^{-1}(0.2074) \] Calculate \(C\): \[ C \approx 77.80^\circ \]

Step 7: Conclusion: Largest angle identified

The largest angle in triangle \(ABC\) is approximately \(77.80^\circ\), which is \(\angle C\).

Key Concepts

Triangle Properties

Understanding triangle properties is crucial for solving problems related to triangles, such as identifying the largest angle in a triangle from given side lengths. Triangles have several key properties:

• The sum of the angles in any triangle is always 180 degrees.
• The sides of a triangle are often denoted as \(a\), \(b\), and \(c\), with corresponding opposite angles \(A\), \(B\), and \(C\).
• The largest angle in a triangle is always opposite the largest side.
• A triangle can be classified based on its sides (equilateral, isosceles, scalene) or its angles (acute, right, obtuse).

In the exercise, we use these properties to identify which angle is the largest based on the side lengths, leading us to the application of the Law of Cosines for further calculations.

Angle Calculation

Calculating angles in a triangle involves using various methods and formulas, depending on the given data. For a triangle with known side lengths, we usually employ the Law of Cosines to find specific angles. To calculate the angle opposite the largest side, we use:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
This formula allows us to find angle \(C\), opposite side \(c\), by rearranging to solve for \(\cos(C)\):

• Calculate the squares of each side.
• Substitute these values into the Law of Cosines equation.
• Simplify and rearrange the equation to isolate \(\cos(C)\).
• Use the inverse cosine function to determine angle \(C\).

Each of these steps helps in systematically calculating the angle, ensuring accuracy in determining triangle properties.

Trigonometric Functions

Trigonometric functions are essential tools for solving problems involving triangles, particularly when determining angles from side lengths. Key trigonometric functions include sine, cosine, and tangent, each relating different sides of a right-angled triangle. In non-right-angled triangles, as seen in the Law of Cosines:

• Cosine (\(\cos\)) helps find angles when the lengths of all sides are known.
• The inverse cosine function (\(\cos^{-1}\)) is used to calculate the angle itself from the cosine value.
• These trigonometric functions are not limited to right triangles; they can also be applied in oblique triangles like triangle \(ABC\).

In the given exercise, we ascertain the angle \(C\) by calculating \(\cos(C)\) and using \(\cos^{-1}\) to find the precise angle, demonstrating the powerful role trigonometric functions play in geometry.