Problem 64 - Trigonometry

Question

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. Find \(B\) for triangle \(A B C\) if \(a=13.8\) yards, \(b=22.3\) yards, and \(c=9.50\) yards. a. \(34.3^{\circ}\) b. \(145.7^{\circ}\) C. \(13.8^{\circ}\) d. \(160^{\circ}\)

Step-by-Step Solution

Answer

The angle \( B \) is \( 145.7^{\circ} \), option b.

Step 1: Identify the Law of Cosines

To find angle \( B \), we use the Law of Cosines: \( b^2 = a^2 + c^2 - 2ac \cos(B) \). This formula relates the sides of a triangle to one of its angles.

Step 2: Substitute the Known Values

Substitute the given side lengths into the equation: \( 22.3^2 = 13.8^2 + 9.5^2 - 2 \cdot 13.8 \cdot 9.5 \cdot \cos(B) \).

Step 3: Calculate the Squared Values

Calculate the squares of the side lengths: \( 22.3^2 = 497.29 \), \( 13.8^2 = 190.44 \), \( 9.5^2 = 90.25 \). Thus, the equation becomes: \( 497.29 = 190.44 + 90.25 - 2 \times 13.8 \times 9.5 \cdot \cos(B) \).

Step 4: Simplify the Equation

Add the squared values of \( a \) and \( c \): \( 190.44 + 90.25 = 280.69 \). Substitute back into the equation: \( 497.29 = 280.69 - 2 \times 13.8 \times 9.5 \cdot \cos(B) \).

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

Rearrange the equation to isolate \( \cos(B) \): \( 2 \times 13.8 \times 9.5 \cdot \cos(B) = 280.69 - 497.29 \). Calculate the right side: \( 280.69 - 497.29 = -216.6 \).

Step 6: Complete the Calculation for \( \cos(B) \)

Divide both sides by \( (2 \cdot 13.8 \cdot 9.5) \): \( \cos(B) = \frac{-216.6}{262.2} \). Compute the division: \( \cos(B) \approx -0.826 \).

Step 7: Find Angle \( B \) Using Inverse Cosine

Use the inverse cosine function to find \( B \): \( B = \cos^{-1}(-0.826) \). Calculate \( B \) to get approximately \( 145.7^{\circ} \).

Key Concepts

Triangle Geometry

In triangle geometry, understanding the relationships between angles and sides is fundamental. A triangle is characterized by its three sides and three angles. There are different types of triangles: equilateral, isosceles, and scalene, each with unique properties.

• **Equilateral Triangle:** All sides are equal, and all angles measure 60 degrees.
• **Isosceles Triangle:** Two sides are equal, with two equal angles opposite those sides.
• **Scalene Triangle:** A triangle with all sides and angles being different.

For scalene triangles such as the one in the problem (with sides 13.8, 22.3, and 9.5 yards), understanding how to apply the Law of Cosines is crucial. This rule helps determine the measure of an unknown angle when all three sides are known, as shown in the exercise.

Angle Calculation

Calculating angles in a triangle follows specific mathematical laws and rules. One of the most versatile tools is the Law of Cosines, especially for non-right triangles or when sides are given.
The equation is: \[ b^2 = a^2 + c^2 - 2ac \cdot \cos(B) \] In this formula:

• \(a, b, c\) are the lengths of the triangle’s sides.
• \(B\) is the angle we aim to find.

The exercise applied this formula by substituting the side lengths into the equation to calculate \(\cos(B)\). Working through calculations step-by-step, each mathematical operation simplifies the expression, isolating the cosine component, and leading towards finding the angle.

Inverse Cosine Function

Once you have the cosine of an angle from your calculations, finding the angle itself involves using the inverse cosine function. This function, \( \cos^{-1} \), helps to 'reverse' the cosine value, thus retrieving the angle.

• **Usage:** Given a cosine value (\(-0.826\) in this problem), use \( \cos^{-1}(-0.826) \) to find the angle \(B\).
• **Result:** This function tells us how many degrees or radians we have in our angle.

In the exercise, computing \( \cos^{-1}(-0.826) \) reveals that angle \( B \) is approximately \( 145.7^{\circ} \). Using a scientific calculator, make sure it is set to degree mode to acquire an answer in degrees. This step is critical for ensuring an accurate result.