Problem 3 - Contemporary Precalculus

Question

Directions: Standard notation for triangle \(A B C\) is used throughout. Use a calculator and round off your answers to one decimal place at the end of the computation. Solve the triangle ABC under the given conditions. $$C=118^{\circ}, a=6, b=12$$

Step-by-Step Solution

Answer

Question: Given a triangle ABC with angle C = 118°, side a = 6, and side b = 12, find the remaining sides and angles of the triangle. Answer: In the triangle ABC, side c is approximately 13.7 units, angle A is approximately 26.2°, and angle B is approximately 35.8°.

Step 1: Finding Angle A Using Law of Sines

The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is constant. It can be written as: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ Since we have angle C, side a, and side b, we can use this formula to find angle A. We have: $$\frac{6}{\sin A} = \frac{12}{\sin(118^\circ)}$$ To solve for angle A, we'll first find the value of \(\sin(118^\circ)\) and then rearrange the equation.

Step 2: Finding the Value of \(\sin(118^\circ)\)

Using a calculator, we find that: $$\sin(118^\circ) \approx 0.875$$

Step 3: Solving for Angle A

Now we can rearrange the Law of Sines equation to isolate angle A on one side: $$\sin A = \frac{6\cdot 0.875}{12}$$ Calculating this, we get: $$\sin A \approx 0.4375$$ Now, to find angle A, we take the inverse sine (or arcsine) of 0.4375: $$A = \arcsin(0.4375)$$ Using a calculator, we find that angle A is approximately: $$A \approx 26.2^\circ$$

Step 4: Finding Angle B

Since the sum of the angles of a triangle is 180 degrees, we can find angle B by subtracting angles A and C from 180 degrees: $$B = 180^\circ - A - C$$ Substituting our known values for A and C, we get: $$B = 180^\circ - 26.2^\circ - 118^\circ$$ Which gives us: $$B \approx 35.8^\circ$$

Step 5: Finding Side c Using Law of Sines

Now we can use the Law of Sines again to find side c. We'll use the following form of the Law of Sines equation: $$\frac{c}{\sin C} = \frac{a}{\sin A}$$ Substituting our known values for A, C, a, and their respective sines, we get: $$\frac{c}{\sin(118^\circ)} = \frac{6}{\sin(26.2^\circ)}$$ Rearranging the equation and substituting the known sine values, we get: $$c = \frac{6\cdot \sin(118^\circ)}{\sin(26.2^\circ)}$$ Calculating this, we find that side c is approximately: $$c \approx 13.7$$ We have now solved the triangle ABC with sides a = 6, b = 12, c = 13.7 and angles A = 26.2º, B = 35.8º, and C = 118º.

Key Concepts

Law of Sines

When solving a triangle, the Law of Sines is an essential tool, especially when dealing with non-right angled triangles. This law allows us to set up a relationship between the sides of a triangle and their opposite angles.

The Law of Sines is particularly useful when you have the following combinations of known values: two angles and one side (AAS or ASA) or two sides and a non-enclosed angle (SSA). In these cases, the Law of Sines is written as:
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\] where 'a', 'b', and 'c' represent the triangle's side lengths, and 'A', 'B', and 'C' represent the corresponding opposite angles.

In the example provided in the original exercise, the Law of Sines is adeptly applied to find the unknown angle 'A'. It's important to note that rounding should be done only at the end of computations to maintain accuracy.

Triangle Angle Calculation

Calculating angles in a triangle is a basic yet vital part of trigonometry and geometry. The sum of the interior angles of any triangle is always 180 degrees. Therefore, knowing just one angle allows you to calculate the others if you have additional information such as side lengths.

The process usually involves knowing at least two angles to find the third. However, in the provided problem, we have one angle and the lengths of two sides, and we use the Law of Sines to find another angle. Once we have two angles, the third angle 'B' is easily found by subtracting the known angles from 180 degrees:
\[B = 180^\circ - A - C\]

Tip For Students:

Always double-check your calculations for angles in a triangle. A small mistake in angle computation can lead to incorrect results for the rest of the problem.

Trigonometric Ratios

Trigonometric ratios are the foundation of trigonometry, relating angles to the sides of a triangle. These ratios include sine (sin), cosine (cos), and tangent (tan), among others.

In the context of solving triangles, these ratios help determine unknown sides or angles. For example, the sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse.

In our example, the sine function is critical in applying the Law of Sines. We used the sine of angle 'C' and the known side 'a' to find the value of the sine of angle 'A', and eventually, through inversing the sine function, we obtained the measure of angle 'A'.

Cautious Computation:

When working with trigonometric functions, it's important to ensure your calculator is set to the correct mode (degrees or radians) to avoid errors.