Problem 13 - Precalculus with Limits

Question

In Exercises 5-20, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. \(A = 120^{\circ}\), \(b = 6\), \(c = 7\)

Step-by-Step Solution

Answer

The values of side a and angles B and C can be calculated using the Law of Cosines. The values are found to be approximately: a = 11.66 units, B = 26.57 degrees and C = 33.43 degrees.

Step 1: Calculate side a using the Law of Cosines

The Law of Cosines states that for any triangle with sides of lengths a, b, and c and with the corresponding angles labeled A, B, and C respectively, the following holds true: \(a^2= b^2 + c^2 - 2bc \cdot cos(A)\). Substituting values into this equation, we have \(a^2 = 6^2 + 7^2 - 2 \cdot 6 \cdot 7 \cdot cos(120^{\circ})\). Solve this equation to find value of a.

Step 2: Calculate angle B using the Law of Cosines

Now, let's solve for angle B using the rearranged Law of Cosines: \(\cos(B) = (a^2 + c^2 - b^2) / 2ac\). Substituting the values available, we have the equation \(\cos(B) = (a^2 + 7^2 - 6^2) / 2a \cdot 7\). Solve this equation to find value of angle B, and don't forget to convert the arc cosine value to degrees.

Step 3: Calculate angle C

Remember that the sum of all angles in a triangle equals 180 degrees. Thus, angle C can be calculated as \(C = 180^{\circ} - A - B\). Substitute the given and calculated values of angles A and B into this equation to find the value of angle C. Remember to round your answer to two decimal places.

Key Concepts

Triangle Solving

Solving a triangle means finding all the unknown sides and angles when certain initial data is given. The Law of Cosines is highly useful in this process, especially when dealing with non-right triangles. The objective here is to determine unknown components based on known values of angles and sides. In our problem, we start with angle \(A\) being \(120^{\circ}\), \(b = 6\), and \(c = 7\). The main task is to calculate the unknown side \(a\) and the two other angles, \(B\) and \(C\). By using math principles such as the Law of Cosines and angle sum in triangles, you can systematically figure out each component. The solving process involves substituting known values into applicable formulas and solving for unknowns. Each calculation builds on the previous, gradually revealing the complete picture of the triangle's dimensions.

Angle Calculation

Calculating angles in a triangle requires careful application of formulas and understanding the properties of triangles. First, we use the Law of Cosines to find angles step-by-step. After solving for side \(a\), the next goal is to find angle \(B\). For this, we rearrange the Law of Cosines formula:

• \(\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}\)

Substitute the known and calculated values, and isolate \(\cos(B)\). This gives the cosine of angle \(B\), which we can then convert to degrees using the inverse cosine function. The calculation of angle \(C\) is straightforward once you have angles \(A\) and \(B\), as they must sum to \(180^{\circ}\). Simply use the formula:

• \(C = 180^{\circ} - A - B\)

This straightforward subtraction gives you the third angle, completing the angle calculation process with ease.

Side Length Calculation

Finding the length of a side when other sides and angles are known involves the direct application of the Law of Cosines. In our example, to find side \(a\), we use the formula:

• \(a^2 = b^2 + c^2 - 2bc \cdot \cos(A)\)

Substitute \(b\) as 6, \(c\) as 7, and \(A\) as \(120^{\circ}\). Calculate \(\cos(120^{\circ})\), which will be negative due to the obtuse angle, affecting the equation by increasing the value of \(a^2\).
Once the formula is applied and solved, take the square root to find \(a\).
The side calculations not only provide insights into the triangle's proportions but also play a key role in verifying the correctness of angle calculations. Each consistency check in the triangle helps solidify understanding and ensure accuracy in all measurements.