Problem 23 - Contemporary Precalculus

Question

Solve the triangle. The Law of Cosines may be needed. $$a=16.5, b=18.2, C=47^{\circ}$$

Step-by-Step Solution

Answer

Question: Solve the triangle with given side lengths a = 16.5, b = 18.2, and angle C = 47° using the Law of Cosines and Law of Sines. Answer: The triangle has side lengths a = 16.5, b = 18.2, c ≈ 25.20 and angle measures A ≈ 33.74°, B ≈ 99.26°, and C = 47°.

Step 1: Apply the Law of Cosines

We start by using the Law of Cosines, which states that $$c^2 = a^2 + b^2 - 2ab * \cos{C}$$ In this problem, we are given $$a=16.5, b=18.2, C=47^{\circ}$$ Plugging these values into the Law of Cosines formula, we get $$c^2 = (16.5)^2 + (18.2)^2 - 2(16.5)(18.2) * \cos{47^{\circ}}$$

Step 2: Calculate c

Now, we can calculate the value of c. By simplifying the formula, we get $$c^2 \approx 635.1592$$ Taking the square root, we find $$c \approx 25.20$$

Step 3: Apply the Law of Sines

Now that we have all three side lengths, we can use the Law of Sines to find angles A and B. The Law of Sines states that $$\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}$$ We will use this formula twice while solving for A and B separately.

Step 4: Calculate angle A

We can find angle A by rearranging the Law of Sines formula to isolate angle A: $$A = \arcsin{\left(\frac{a * \sin{C}}{c}\right)}$$ Plugging in the values, we get $$A = \arcsin{\left(\frac{16.5 * \sin{47^{\circ}}}{25.20}\right)}$$ Calculating this, we find $$A \approx 33.74^{\circ}$$

Step 5: Calculate angle B

Now that we have A and C, we can find angle B by knowing that all three angles in a triangle must add up to 180 degrees. Therefore, $$B = 180^{\circ} - A - C$$ Substituting the values, we get $$B = 180^{\circ} - 33.74^{\circ} - 47^{\circ}$$ Calculating this, we find $$B \approx 99.26^{\circ}$$

Step 6: Final Answer

Now we have all the side lengths and angle measures for the triangle: $$a=16.5, b=18.2, c=25.20$$ $$A=33.74^{\circ}, B=99.26^{\circ}, C=47^{\circ}$$

Key Concepts

Triangle Solving

Solving a triangle involves finding all unknown sides and angles of the triangle using given information. In this particular problem, we were given two sides and one angle, known commonly as the SAS (Side-Angle-Side) scenario. Using this information, the goal is to determine all the missing measurements. Here’s how it works in general:

You begin with the given values:

• Sides: \(a = 16.5\) and \(b = 18.2\)
• Angle: \(C = 47^{\circ}\)

By employing the Law of Cosines, you find the third side. After determining all sides, the next step is to use the Law of Sines to find the remaining angles. Understanding this step-by-step approach ensures you grasp each stage of triangle solving fully. After following the process, all side lengths and angles are known, making your triangle fully "solved."

Law of Sines

The Law of Sines is a fundamental tool in trigonometry, especially useful in finding unknown angles or sides in a triangle when certain other measures are known. In essence, it states that the ratio of the length of a side to the sine of its opposite angle is the same for all three sides and angles in a triangle. The formula is:\[ \frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c} \]
In our example, after determining the third side using the Law of Cosines, we then applied the Law of Sines to find each of the missing angles. First, it helped us calculate angle \(A\) using:\[ A = \arcsin\left(\frac{a \times \sin{C}}{c}\right) \]After determining \(A\), the same principle allows us to solve for angle \(B\), knowing each step ensures accuracy and clarity in finding the triangle's angles.

Triangle Angles

Understanding the properties of triangle angles is crucial in solving any triangle. A critical feature is that the sum of a triangle's interior angles is always \(180^{\circ}\). This rule is immensely helpful when calculating the final angle, once two angles are known. For instance, in this exercise:

• We found angle \(A\) using the Law of Sines.
• The known angle \(C\) was already provided.

Therefore, finding angle \(B\) becomes a simple subtraction problem using the formula: \[ B = 180^{\circ} - A - C \]This calculation ensures all angles correctly add up to \(180^{\circ}\), validating your results as a correctly solved triangle. Keeping this in mind simplifies the approach to every triangle problem you encounter.