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Chapter 6: Problem 99

Solve the triangle in which \(a=5, b=9\), and \(c=12\).

### Short Answer

Expert verified

C ≈ 115.08°, A ≈ 22.48°, B ≈ 42.44°

## Step by step solution

01

## Use the Law of Cosines to Find Angle C

Use the Law of Cosines to find the largest angle first, which is opposite the longest side \( c \). Formula: \[ c^2 = a^2 + b^2 - 2ab\cos(C) \]Substitute the known values:\[ 12^2 = 5^2 + 9^2 - 2 \cdot 5 \cdot 9 \cdot \cos(C) \]Calculate each term: \[ 144 = 25 + 81 - 90 \cos(C) \]Combine like terms: \[ 144 = 106 - 90 \cos(C) \]Isolate \( \cos(C) \):\[ 144 - 106 = -90 \cos(C) \]\[ 38 = -90 \cos(C) \]Divide both sides by \(-90\):\[ \cos(C) = -\frac{38}{90} \]Calculate \( \cos(C) \):\[ \cos(C) \approx -0.4222 \]Use the inverse cosine function to find \( C \):\[ C \approx \cos^{-1}(-0.4222) \approx 115.08° \]

02

## Use the Law of Cosines to Find Angle A or B

Now, use the Law of Cosines to find another angle. For example, we can find angle A.Formula: \[ a^2 = b^2 + c^2 - 2bc\cos(A) \]Substitute the known values: \[ 5^2 = 9^2 + 12^2 - 2 \cdot 9 \cdot 12 \cos(A) \]Calculate each term:\[ 25 = 81 + 144 - 216 \cos(A) \]Combine like terms:\[ 25 = 225 - 216 \cos(A) \]Isolate \( \cos(A) \):\[ 25 - 225 = -216 \cos(A) \]\[ -200 = -216 \cos(A) \]Divide both sides by \(-216\):\[ \cos(A) = \frac{200}{216} \approx 0.9259 \]Use the inverse cosine function to find \( A \):\[ A \approx \cos^{-1}(0.9259) \approx 22.48° \]

03

## Find the Remaining Angle (Angle B)

Use the triangle sum property, which states that the sum of the angles in a triangle is always 180°:\[ B = 180° - A - C \]Substitute the known values:\[ B = 180° - 22.48° - 115.08° \]Calculate \( B \):\[ B \approx 42.44° \]

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### Law of Cosines

The Law of Cosines is a crucial formula for solving triangles, especially when you have the lengths of all three sides. This law relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is:

\[ c^2 = a^2 + b^2 - 2ab\text{cos}(C) \]

In our problem, we use it first to find the largest angle, which is opposite the longest side, length c. By substituting the given values, we solve for \( \cos(C) \). The calculated value of \( \cos(C) \) is then used to find the measure of angle C. This gives us a good starting point to determine the remaining angles.

###### Angle Calculation

Once you have found one angle using the Law of Cosines, you can use it to find the other angles. In the exercise, we started by calculating angle C. Knowing one angle helps simplify finding the remaining angles. For angle A, we use the same Law of Cosines but tweak the sides involved:

\[ a^2 = b^2 + c^2 - 2bc \text{cos}(A) \]

Iterating through similar steps, we substitute the values and isolate \( \text{cos}(A) \). This progressive method ensures accurate and organized calculations for each angle. The main focus while finding angles is to always use the correct values and double-check each step.

###### Triangle Sum Property

The triangle sum property is a straightforward yet powerful tool. According to this property, the sum of the interior angles in any triangle is always \(180^\text{°}\). After finding two angles, the third angle can be quickly calculated by subtracting the sum of the known angles from \(180^\text{°}\):

\[ B = 180^\text{°} - A - C \]

This method leverages simple subtraction to provide the remaining angle, ensuring no complex calculations are needed once the first two angles are known. In the exercise, after calculating angles A and C, finding angle B became a straightforward subtraction problem.

###### Inverse Cosine Function

The inverse cosine function, denoted as \( \text{cos}^{-1} \), is essential for finding angles from cosine values. When using the Law of Cosines, you often end up with a cosine value that needs to be converted into an angle. The notation \( \text{cos}^{-1}(x) \) helps us find the angle whose cosine is x. Here's how it applies:

If \( \text{cos}(C) = -0.4222 \), then \( C = \text{cos}^{-1}(-0.4222) \), which gives us an angle measure. Technology such as calculators or mathematical software makes this step easy by providing the angle directly when the inverse cosine function is used. This function bridges the values obtained through the Law of Cosines to actual angle measures needed for the triangle.

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