Problem 51 - Algebra 2

Question

Solve each equation. Round to the nearest tenth. $$ 7^{2}=11^{2}+9^{2}-2(11)(9) \cos B^{\circ} $$

Step-by-Step Solution

Answer

Angle \(B\) is approximately \(39.4^\circ\).

Step 1: Understand the Equation

The given equation is based on the Law of Cosines: \(c^2 = a^2 + b^2 - 2ab \cos C\). Here, we have \(7^2 = 11^2 + 9^2 - 2(11)(9) \cos B\). Our goal is to solve for \(\cos B\).

Step 2: Calculate the Squares of Known Values

Calculate \(7^2\), \(11^2\), and \(9^2\):- \(7^2 = 49\)- \(11^2 = 121\)- \(9^2 = 81\)

Step 3: Substitute and Simplify

Substitute the squares back into the equation: \(49 = 121 + 81 - 2(11)(9) \cos B\).Simplify the right side: \(49 = 202 - 198 \cos B\).

Step 4: Isolate \(\cos B\)

Move 202 from the right side to the left by subtracting: \(49 - 202 = -198 \cos B\).This simplifies to:\(-153 = -198 \cos B\).

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

Divide both sides by \(-198\) to find \(\cos B\):\(\cos B = \frac{-153}{-198}\).Calculate:\(\cos B \approx 0.7727\).

Step 6: Find Angle \(B\)

Use the inverse cosine function to find \(B\):\(B \approx \cos^{-1}(0.7727)\).Calculate the angle:\(B \approx 39.4^\circ\) (rounded to the nearest tenth).

Key Concepts

Inverse Cosine

In trigonometry, the inverse cosine function is crucial for finding angles when we know the cosine value. This is often needed in real-life applications where we use the Law of Cosines. The inverse cosine, also known as arccosine, is denoted as \( \cos^{-1} \). It works by telling us which angle, between 0\( ^\circ \) and 180\( ^\circ \), would have a specified cosine value.

To use \( \cos^{-1} \), enter the value of the cosine you know into a calculator that has the inverse cosine function. For example, if \( \cos B = 0.7727 \), then \( B \) can be found as follows: \( B \approx \cos^{-1}(0.7727) \).

This calculator step gives the angle projection in degrees, making it simple to translate from cosine value to the angle itself. This is especially useful when using the Law of Cosines, as it allows us to find unknown angles in a triangle if we know the side lengths.

Trigonometric Equations

Trigonometric equations involve relationships between side lengths and angles in triangles. The Law of Cosines provides a helpful equation for this: \( c^2 = a^2 + b^2 - 2ab \cos C \).

The Law of Cosines is particularly useful in solving for angles in non-right triangles where direct measurement isn't possible. In our problem, we substitute values for known sides into this equation to find the unknown \( \cos \, B \). These equations usually require patience in simplifying and manipulating terms.

• Start by understanding the structure: \( a^2 + b^2 - 2ab \cos C \) where \( a \), \( b \), and \( c \) are the side lengths of the triangle.
• Solve for unknowns systematically by isolating variables.
• Simplify each part carefully to ensure accuracy.

After finding \( \cos B \), use the inverse cosine to find the angle. To master solving these equations, practice by working through similar problems.

Angle Measurement

Finding the exact angle in a triangle when using trigonometric solutions is a critical step in interpreting real-world problems. Angles can be measured in degrees or radians, but degrees are more common in elementary trigonometry due to their straightforward nature.

In our exercise, after calculating \( \cos B \), we needed to determine the measure of angle \( B \) using the inverse cosine function. Angle measurements involve essential practices like rounding. Here, angle \( B \approx 39.4^\circ \) was rounded to the nearest tenth to provide a clear, concise measure.

• Ensure calculator is set to the correct mode (degrees or radians).
• Input calculations carefully to avoid rounding errors.
• Understand rounding rules when a specific precision is required.

By following these steps, we can swiftly measure angles during calculations. Mastery of angle measurement is vital across fields, especially when precise calculations are needed.