Problem 11 - 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. $$a=16, b=30, c=32$$

Step-by-Step Solution

Answer

Question: Given a triangle ABC with sides a = 16, b = 30, and c = 32, find the angles A, B, and C. Answer: In triangle ABC, the angles are A = 30.0°, B = 68.2°, and C = 81.8°.

Step 1: Understand the Law of Cosines

The Law of Cosines states that for any triangle with sides a, b, and c, and respective angles A, B, and C: $$a^2 = b^2 + c^2 - 2bc\cos{A}$$ $$b^2 = a^2 + c^2 - 2ac\cos{B}$$ $$c^2 = a^2 + b^2 - 2ab\cos{C}$$ We will use this formula to find the angles A, B, and C in triangle ABC.

Step 2: Calculate Angle A

Using the Law of Cosines, we can find angle A, using the formula: $$a^2 = b^2 + c^2 - 2bc\cos{A}$$ Substitute the values of a, b, and c: $$16^2 = 30^2 + 32^2 - 2(30)(32)\cos{A}$$ Now we will solve for $$\cos{A}$$: $$\cos{A} = \frac{30^2 + 32^2 - 16^2}{2(30)(32)}$$ Calculate the value: $$\cos{A} = \frac{900 + 1024 - 256}{1920} = \frac{1668}{1920} = 0.869$$ Now, find angle A using the inverse cosine function: $$A = \cos^{-1}(0.869) = 30.0°$$

Step 3: Calculate Angle B

Similarly, use the Law of Cosines for angle B: $$b^2 = a^2 + c^2 - 2ac\cos{B}$$ Substitute values and solve for $$\cos{B}$$: $$\cos{B} = \frac{16^2 + 32^2 - 30^2}{2(16)(32)}$$ Calculate the value: $$\cos{B} = \frac{256 + 1024 - 900}{1024} = \frac{380}{1024} = 0.371$$ Now, find angle B using the inverse cosine function: $$B = \cos^{-1}(0.371) = 68.2°$$

Step 4: Calculate Angle C

Since we know that the sum of the angles in a triangle is 180 degrees, we can simply find angle C by subtracting angles A and B from 180°: $$C = 180° - A - B$$ Substitute values: $$C = 180° - 30.0° - 68.2°$$ Calculate the value: $$C = 81.8°$$

Step 5: Present the Solution

In triangle ABC with sides a = 16, b = 30, and c = 32, the angles are: $$A = 30.0°$$ $$B = 68.2°$$ $$C = 81.8°$$

Key Concepts

Solving Triangles

Solving triangles is a fundamental concept in trigonometry, involving finding the unknown sides and angles of a triangle. In the given exercise, we are provided with a non-right triangle with side lengths of a=16, b=30, and c=32. To solve this triangle, we use the Law of Cosines, a powerful tool that relates the lengths of a triangle's sides to its angles.

When two sides and the included angle are known, or all three sides are known—like in our case—the Law of Cosines becomes particularly useful. The method involves substituting the known side lengths into the law's formula to find one angle, then using that knowledge in conjunction with other trigonometric principles to find the remaining angles and potentially the other sides.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. In a right-angled triangle, these functions are sine, cosine, and tangent, but they can also be applied to non-right triangles through the Law of Cosines and Sines.

In our problem, the cosine function is central to calculating the angles. The cosine of an angle in a triangle is the ratio of the length of the adjacent side to the length of the hypotenuse (the side opposite the right angle). However, in non-right triangles, we use the Law of Cosines, which generalizes the cosine function to relate all three sides of the triangle to one of its angles.

Inverse Cosine Function

The inverse cosine function, denoted as cos⁻¹ or arccos, is used to find the angle when the cosine value is known. It's the reverse process of finding the cosine of an angle; instead, we are given the cosine of an angle and we need to determine the angle itself.

The range of the inverse cosine function is from 0 to 180 degrees, making it perfect for finding angles in triangles. In our exercise, after calculating the cosine of angle A to be 0.869 using the Law of Cosines, we use the inverse cosine function to find that the angle A itself is 30.0 degrees.

Angle Calculation

Calculating angles in a triangle can be done through various methods, depending on the given information. In the context of our exercise, once one angle of the triangle is found using the Law of Cosines and the inverse cosine function, the remaining angles can be calculated using different strategies.

For instance, after finding angle A, we could apply the Law of Cosines again to find angle B. Lastly, the angle C can be inferred since the sum of all angles in a triangle is always 180 degrees. By subtracting the known angles A and B from 180 degrees, we obtain the measure of angle C.