Problem 26 - Contemporary Precalculus

Question

Solve the triangle. The Law of Cosines may be needed. $$b=24.1, c=10.5, C=26.3^{\circ}$$

Step-by-Step Solution

Answer

Answer: The approximate length of side a is 15.44 units, Angle A is approximately 43.5 degrees, and Angle B is approximately 110.2 degrees.

Step 1: Apply the Law of Cosines to find side a

To find side a, we can use the Law of Cosines, which is given by: $$a^2 = b^2 + c^2 - 2bc\cos{C}$$ In this case, we were given: $$b = 24.1, c = 10.5, C = 26.3^{\circ}$$ Plug the given values into the Law of Cosines formula: $$a^2 = 24.1^2 + 10.5^2 - 2(24.1)(10.5)\cos{26.3^{\circ}}$$

Step 2: Calculate side a

Solve the equation to find the length of side a: $$a^2 = 580.81 + 110.25 - 506.1\cos{26.3^{\circ}}$$ $$a^2 \approx 691.06 - 452.56$$ $$a^2 \approx 238.50$$ $$a \approx \sqrt{238.50} \approx 15.44$$ So, side a is approximately 15.44 units in length.

Step 3: Apply the Law of Sines to find angle A

Now that we have all three side lengths, we can use the Law of Sines to find angle A. The Law of Sines states: $$\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}$$ Using the information we have for angle C and sides a and c, we can set up an equation to find angle A: $$\frac{\sin{A}}{15.44} = \frac{\sin{26.3^{\circ}}}{10.5}$$ To find \(\sin{A}\), cross-multiply: $$\sin{A} \approx \frac{10.5\sin{26.3^{\circ}}}{15.44}$$ And then find angle A: $$A \approx \arcsin{\left(\frac{10.5\sin{26.3^{\circ}}}{15.44}\right)} \approx 43.5^{\circ}$$ So, angle A is approximately 43.5 degrees.

Step 4: Calculate angle B

Since the sum of the angles in a triangle is always 180 degrees, we can now find angle B by subtracting the other two known angles from 180: $$B = 180^{\circ} - A - C$$ $$B \approx 180^{\circ} - 43.5^{\circ} - 26.3^{\circ}$$ $$B \approx 110.2^{\circ}$$ So, angle B is approximately 110.2 degrees. In conclusion, the unknown side a is approximately 15.44 units, Angle A is approximately 43.5 degrees, and Angle B is approximately 110.2 degrees.

Key Concepts

Solving Triangles

Solving triangles is the process of finding the unknown sides and angles of a triangle when we are given some of its measurements. Generally, we might have the measurements of two sides and an angle (SAS), two angles and a side (ASA or AAS), or all three sides (SSS). Various strategies, including the Law of Cosines and the Law of Sines, can be employed depending on the information given.

To solve a triangle, we typically start with the Law of Cosines if we're given SAS or SSS, which helps us find a missing side. With the side(s) known, we can then switch to the Law of Sines to find unknown angles. It's crucial to know which formula to apply to avoid confusion and errors in calculation.

Law of Sines

The Law of Sines is a formula that relates the sides of a triangle to the sine of its opposite angles. It's written as:
\[\begin{equation}\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}\end{equation}\]where 'A', 'B', and 'C' are the angles, and 'a', 'b', and 'c' are the sides opposite these angles respectively.

The Law of Sines is particularly useful when dealing with ASA or AAS cases, allowing us to find unknown sides or angles. When an angle and its opposite side are known, you can set up a proportion to solve for the missing measurements. Remember that the Law of Sines works well for triangles without a right angle and where at least one angle-side opposite pair is known.

Trigonometric Equations

Trigonometric equations involve trigonometric functions, such as sine, cosine, and tangent, and these can be used to find the unknown values in a triangle. For instance, the Law of Cosines and Law of Sines give rise to trigonometric equations which we can solve for a specific variable.

When solving, we sometimes rearrange these equations, take inverse trigonometric functions, or apply other algebraic methods to isolate and solve for the unknowns. It's essential to be comfortable with manipulating and solving these equations, as they are at the heart of trigonometry.

Triangle Angle Calculation

Triangle angle calculation involves determining the measures of the angles within a triangle. Knowing that the sum of the angles in any triangle is 180 degrees, one can easily calculate a missing angle once the others are known.

In more complex cases, such as non-right triangles without angle measurements, we may need to employ the Law of Sines or Law of Cosines to first find a side or an angle, and then deduce the others. Accurate triangle angle calculation is fundamental in not only solving triangles but also in various applications in science, engineering, and other fields. Patience and precision in calculation are key in avoiding errors, especially when working through multiple steps.