Problem 17 - Precalculus with Limits

Question

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ A=120^{\circ}, \quad b=6, \quad c=7 $$

Step-by-Step Solution

Answer

The length of side a is approximately 11.48 units. The measures of angles B and C are approximately 39.13 and 20.87 degrees, respectively.

Step 1: Find the length of side a

Using the Law of Cosines, substitute the given lengths and angle into the formula to solve for side a. In specific, the formula is \(a = \sqrt{b^2 + c^2 - 2bc \cos A}\), where: \(A = 120^{\circ}, \quad b = 6, \quad c = 7\). After calculating, you should get: \(a = \sqrt{6^2 + 7^2 - 2(6)(7) \cos(120)}\), which simplifies to \(a \approx 11.48\). So, the length of side a is approximately 11.48 units.

Step 2: Solve for Angle B

Using the Law of Cosines again, rearrange the formula to solve for angle B. In specific, the formula is \(\cos B = (a^2 + c^2 - b^2) / 2ac\). Substituting the values of sides a, b, and c, you get: \(\cos B = (11.48^2 + 7^2 - 6^2) / 2 * 11.48 * 7\). After calculating, you should get: \(B = \cos^{-1}[(11.48^2 + 7^2 - 6^2) / 2 * 11.48 * 7] = 39.13^{\circ}\). So, the measure of angle B is approximately 39.13 degrees.

Step 3: Solve for Angle C

We can find angle C by subtracting the measures of angles A and B from 180°, because the sum of the measures in a triangle is 180°. So, \(C = 180^{\circ} - A - B = 180^{\circ} - 120^{\circ} - 39.13^{\circ}\). So, the measure of angle C is approximately 20.87 degrees.

Key Concepts

Triangle Solution

Solving a triangle means identifying all its sides and angles. In the scenario given, we know one angle, which is \( 120^{\circ} \), and two sides, \( b = 6 \) and \( c = 7 \). To fully solve the triangle, we need to find the missing side, \( a \), and the other two angles, \( B \) and \( C \).

The Law of Cosines is especially helpful in these cases, where you have two sides and the included angle or all three sides. This law relates the lengths of the sides of a triangle to the cosine of one of its angles. Here, it allows us to calculate the length of side \( a \).

The formula \( a = \sqrt{b^2 + c^2 - 2bc \cos A} \) is used to find the third side of the triangle. After plugging in the known values:\( A = 120^{\circ}, b = 6, \) and \( c = 7 \), we calculate \( a \approx 11.48 \).

Once all sides are known, remaining angles can be calculated using the Law of Cosines again or by using simpler trigonometry methods such as subtracting known angles from 180° to find the final unknown angle.

Angle Calculation

After determining the length of side \( a \), the next step is to calculate the angles \( B \) and \( C \). The angle \( B \) can be found using the rearranged Law of Cosines. This is needed since all three side lengths are now known. The formula used is \( \cos B = \left( \frac{a^2 + c^2 - b^2}{2ac} \right) \).

By substituting the values \( a = 11.48, b = 6, \) and \( c = 7 \), we can calculate the angle \( B \approx 39.13^{\circ} \). This solution arises from reversing the cosine function using the arccosine or \( \cos^{-1} \) function to find the angle measure.

With angles \( A \) and \( B \) known, angle \( C \) can be deduced by subtracting the sum of angles \( A \) and \( B \) from the total of 180°: \( C = 180^{\circ} - 120^{\circ} - 39.13^{\circ} \), which gives \( C \approx 20.87^{\circ} \). This is derived from the fact that the sum of angles in any triangle is always 180°.

Trigonometry Concepts

Trigonometry revolves around the relationship between angles and sides in triangles. Key to solving triangle problems like this is understanding the core trigonometric functions and laws. The Law of Cosines is a critical concept for non-right triangles, allowing us to solve for unknown elements when three sides or two sides and an included angle are known.

Trigonometric principles include:-

• Law of Cosines: Connects angles to side lengths in non-right triangles.
• Cosine Function: Represents the ratio of adjacent side over hypotenuse in right triangles and helps find angles when side lengths are known.
• Sum of Angles: The angles inside a triangle always add up to 180°.

These principles provide a structure to solve complex triangle problems, integrating angles and side lengths in real-world and theoretical contexts. With practice, these concepts become intuitive tools for analyzing triangles.