Problem 6 - Precalculus with Calculus Previews

Question

Use the Law of Cosines to solve the triangle. $$ a=7, b=9, c=4 $$

Step-by-Step Solution

Answer

The triangle's angles are approximately: \( A = 48.2^\circ \), \( B = 106.3^\circ \), \( C = 25.5^\circ \).

Step 1: Understanding the Problem

We are given a triangle with side lengths: \( a = 7 \), \( b = 9 \), and \( c = 4 \). We need to find the measures of the triangle’s angles using the Law of Cosines.

Step 2: Applying Law of Cosines to Find Angle A

To find angle \( A \), we use the Law of Cosines formula: \[ \cos A = \frac{{b^2 + c^2 - a^2}}{{2bc}} \]Substitute the given values: \[ \cos A = \frac{{9^2 + 4^2 - 7^2}}{{2 \times 9 \times 4}} \] Calculate this to find \( \cos A \).

Step 3: Calculating \( \cos A \)

Substitute and simplify:\[ \cos A = \frac{{81 + 16 - 49}}{{72}} = \frac{48}{72} = \frac{2}{3} \]Use the inverse cosine to find \( A \): \( A = \cos^{-1} \left( \frac{2}{3} \right) \).

Step 4: Applying Law of Cosines to Find Angle B

To find angle \( B \), use:\[ \cos B = \frac{{a^2 + c^2 - b^2}}{{2ac}}\]Substitute the given values:\[ \cos B = \frac{{7^2 + 4^2 - 9^2}}{{2 \times 7 \times 4}}\]Calculate to find \( \cos B \).

Step 5: Calculating \( \cos B \)

Substitute and simplify:\[ \cos B = \frac{{49 + 16 - 81}}{{56}} = \frac{-16}{56} = -\frac{2}{7} \]Use the inverse cosine to find \( B \): \( B = \cos^{-1} \left( -\frac{2}{7} \right) \).

Step 6: Applying Law of Cosines to Find Angle C

Solve for angle \( C \) using the formula:\[ \cos C = \frac{{a^2 + b^2 - c^2}}{{2ab}}\]Substitute in the values:\[ \cos C = \frac{{7^2 + 9^2 - 4^2}}{{2 \times 7 \times 9}}\]Calculate to find \( \cos C \).

Step 7: Calculating \( \cos C \)

Substitute and simplify:\[ \cos C = \frac{{49 + 81 - 16}}{{126}} = \frac{114}{126} = \frac{19}{21} \]Use the inverse cosine to find \( C \): \( C = \cos^{-1} \left( \frac{19}{21} \right) \).

Key Concepts

triangle solving

When solving a triangle, we're tasked with finding unknown sides and angles given certain known values. In our case, we have a triangle where all three side lengths are known: \( a = 7 \), \( b = 9 \), and \( c = 4 \). The process involves determining the measures of all three angles using these side lengths.

This specific problem is an example of solving using the Law of Cosines. It's particularly useful when you know all the sides or two sides and the included angle. It allows you to find the unknown parts of the triangle in an efficient manner.

• Always start by identifying known quantities.
• Apply relevant formulas, like the Law of Cosines, to find missing angles.
• Ensure calculations are precise to get accurate angle measurements.

trigonometry

Trigonometry provides the mathematical framework to analyze triangles, particularly right-angled and oblique triangles. The Law of Cosines is a critical trigonometric tool that extends understanding beyond right triangles. It relates the sides of a triangle to the cosine of one of its angles, helping to solve any type of triangle.

The Law of Cosines is expressed as: \[ c^2 = a^2 + b^2 - 2ab \, \cos C\]Alternatively, for angle calculations:

• To find \( \cos A \): \( \cos A = \frac{{b^2 + c^2 - a^2}}{{2bc}} \)
• To find \( \cos B \): \( \cos B = \frac{{a^2 + c^2 - b^2}}{{2ac}} \)
• To find \( \cos C \): \( \cos C = \frac{{a^2 + b^2 - c^2}}{{2ab}} \)

This method forms a bridge between side length information and angle determination, demonstrating the power of trigonometry in solving triangle problems.

angle calculation

Calculating angles is crucial in triangle problems, particularly when you have all sides known. After using the Law of Cosines to get expressions for \( \cos A \), \( \cos B \), and \( \cos C \), you use inverse trigonometric functions to find the actual angle measures.

• For \( A \), substitute \( \cos A = \frac{2}{3} \) into \( A = \cos^{-1} \left( \frac{2}{3} \right) \).
• For \( B \), substitute \( \cos B = -\frac{2}{7} \) into \( B = \cos^{-1} \left( -\frac{2}{7} \right) \).
• For \( C \), substitute \( \cos C = \frac{19}{21} \) into \( C = \cos^{-1} \left( \frac{19}{21} \right) \).

Inverse cosine (\( \cos^{-1} \)) is used to determine angle values from their cosine representations. Careful handling of negative values, as seen in \( \cos B \), is important, since the range of the inverse cosine function must be fully understood. Employ these calculations to derive angles accurately, paving the way for complete triangle solutions.