Problem 15 - Algebra and Trigonometry

Question

Solve triangle \(A B C\). \(a=20, \quad b=25, \quad c=22\)

Step-by-Step Solution

Answer

The angles are \(A \approx 57.07^\circ\), \(B \approx 49.07^\circ\), and \(C \approx 73.86^\circ\).

Step 1: Confirm Triangle Validity

To ensure the given measurements form a triangle, use the triangle inequality rule. Verify if the sum of any two sides is greater than the third side. Check:- Is \(a + b > c\)? Yes, \(20 + 25 = 45 > 22\).- Is \(a + c > b\)? Yes, \(20 + 22 = 42 > 25\).- Is \(b + c > a\)? Yes, \(25 + 22 = 47 > 20\).Since all these conditions are satisfied, these sides form a valid triangle.

Step 2: Use the Law of Cosines to Find Angle A

Apply the Law of Cosines to find angle \(A\):\[ c^2 = a^2 + b^2 - 2ab \cdot \cos C \]\[ 22^2 = 20^2 + 25^2 - 2 \cdot 20 \cdot 25 \cdot \cos A \]Simplify:\[ 484 = 400 + 625 - 1000 \cdot \cos A \]\[ 484 = 1025 - 1000 \cdot \cos A \]\[ 1000 \cdot \cos A = 1025 - 484 \]\[ 1000 \cdot \cos A = 541 \]\[ \cos A = \frac{541}{1000} \]Calculate \(A\) using the inverse cosine function:\( A = \cos^{-1}(0.541) \approx 57.07^\circ \).

Step 3: Use the Law of Cosines to Find Angle B

Next, use the Law of Cosines to find angle \(B\):\[ a^2 = b^2 + c^2 - 2bc \cdot \cos B \]\[ 20^2 = 25^2 + 22^2 - 2 \cdot 25 \cdot 22 \cdot \cos B \]Simplify:\[ 400 = 625 + 484 - 1100 \cdot \cos B \]\[ 400 = 1109 - 1100 \cdot \cos B \]\[ 1100 \cdot \cos B = 1109 - 400 \]\[ 1100 \cdot \cos B = 709 \]\[ \cos B = \frac{709}{1100} \]Calculate \(B\) using the inverse cosine function:\( B = \cos^{-1}(0.6445) \approx 49.07^\circ \).

Step 4: Calculate Angle C Using the Sum of Angles in a Triangle

Since we have found angles \(A\) and \(B\), we can find angle \(C\) using the fact that the sum of angles in a triangle is \(180^\circ\):\[ C = 180^\circ - A - B \]\[ C = 180^\circ - 57.07^\circ - 49.07^\circ \]\[ C = 73.86^\circ \].

Step 5: Verify Results

Verify the calculated angles' sum to ensure correctness:\[ A + B + C = 57.07^\circ + 49.07^\circ + 73.86^\circ = 180^\circ \].The results are consistent; all calculations confirm a total of \(180^\circ\).

Key Concepts

Triangle Inequality

The Triangle Inequality theorem is a fundamental rule in geometry. It helps determine if three side lengths can form a triangle. According to this theorem:

• The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In our problem with triangle ABC, we had sides with lengths 20, 25, and 22. To see if these lengths form a triangle, we looked at three conditions:

• Check if \(a + b > c\): 20 + 25 = 45, which is indeed greater than 22.
• Check if \(a + c > b\): 20 + 22 = 42, which is greater than 25.
• Check if \(b + c > a\): 25 + 22 = 47, which is greater than 20.

Since all these conditions are satisfied, the side lengths do follow the Triangle Inequality rule and thus can form a triangle.

Angle Calculation

When working with triangles, calculating the angles is an essential step to fully solve the triangle. We often employ the Law of Cosines to do this. The Law of Cosines relates the sides of a triangle to one of its angles. The formula used is:\[c^2 = a^2 + b^2 - 2ab \cdot \cos C\]
From the given side lengths, we can use this formula to solve for angle \(A\), taking side \(c\) as opposite to angle \(A\). We rearranged the formula to solve for the cosine of the angle:\[\cos A = \frac{a^2 + b^2 - c^2}{2ab}\]
After substituting the values of the sides, we calculated \(\cos A\) and found the angle using the inverse cosine function. **Note**: Every triangle has the sum of its three angles equal to \(180^\circ\). After calculating two angles, you can always use this property to find the third angle efficiently.

Inverse Cosine Function

The inverse cosine function, also known as \(\cos^{-1}\), is used to find the angle whose cosine is a given number. It is a crucial aspect of angle calculation in triangles when using the Law of Cosines.To determine angle \(A\) in triangle ABC, we used the result from the law of cosines:\[\cos A = \frac{541}{1000}\]
Using the inverse cosine function, we calculated:\[A = \cos^{-1}(0.541) \approx 57.07^\circ\]
The inverse cosine essentially "undoes" the cosine to give us the angle in degrees. This function is very practical in trigonometry as it converts between trigonometric ratios and angles in a triangle. Always make sure your calculator is set to the correct mode (degrees or radians) when you are performing these calculations.