Problem 10 - Precalculus with Limits

Question

In Exercises 5-20, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. \(a = 55\), \(b = 25\), \(c = 72\)

Step-by-Step Solution

Answer

After following these steps, one will have the measurements of all three angles for the triangle, rounded to two decimal places. It's important to ensure that the sum of all internal angles of the triangle equals 180 degrees.

Step 1: Calculate angle A

The Law of Cosines, when we want to find an angle, is written as: \(cos(A) = [b^2 + c^2 - a^2] / 2bc\). Substitute the given values into the formula: \(cos(A) = [(25)^2 + (72)^2 - (55)^2] / 2 * 25 * 72\). Evaluate the right hand side and use a calculator to find the cosine inverse to get angle A.

Step 2: Calculate angle B

Now use the law of cosines to find angle B: \(cos(B) = [a^2 + c^2 - b^2] / 2ac\). Substitute the given values into the formula: \(cos(B) = [(55)^2 + (72)^2 - (25)^2] / 2 * 55 * 72\). Evaluate the right hand side and use a calculator to find the cosine inverse to get angle B.

Step 3: Calculate angle C

The internal angles in a triangle sum to 180 degrees. Therefore, we will subtract the measures of angles A and B from 180 to get angle C: C = 180 - A - B.

Key Concepts

Solving Triangles

Solving triangles involves finding the missing lengths and angles of a triangle when some measurements are known. It's a fundamental problem in geometry that can be approached in several different ways, depending on the given information. The Law of Cosines is a vital tool for solving triangles, especially when you're dealing with non-right angled triangles where the Pythagorean theorem does not apply. This law relates the lengths of the sides of a triangle to the cosine of one of its angles.
When using the Law of Cosines to solve for an angle, you rearrange the formula to isolate the cosine of the unknown angle. Then, inserting the known side lengths allows you to solve for the cosine of the angle. Lastly, to find the measure of the angle itself, you use the inverse cosine function. It's worth noting that rounding should be done carefully, as premature rounding can lead to significant errors in final calculations. The principles of solving triangles are widely applicable, from navigation and astronomy to designing and architecture.

Trigonometric Functions

Trigonometric functions are a cornerstone of geometry and calculus, providing the relationship between the angles and sides of triangles. These functions include sine, cosine, and tangent, among others. They are defined initially for acute angles in a right triangle as ratios of sides and then extended to angles greater than 90 degrees (obtuse angles) through the unit circle.
In the context of the Law of Cosines, we use the cosine function to express the relationship of an angle to the adjacent side and the opposite side. The cosine of an angle in a triangle is equal to the adjacent side divided by the hypotenuse in a right triangle. However, in any triangle, the Law of Cosines allows us to use cosine for calculations involving any angle and its opposing side. Understanding how to manipulate these functions algebraically and interpret them graphically is fundamental for solving complex problems in trigonometry.

Inverse Cosine

Inverse cosine, often denoted as \(\cos^{-1}\) or \(\arccos\), is a function that 'reverses' the action of the cosine function. It allows you to find the angle that corresponds to a given cosine value. This function is crucial when working with the Law of Cosines because after calculating the cosine of an angle using the formula, you need to take the inverse cosine to find the angle itself.
The range of the inverse cosine function is from 0 to \(\pi\) radians (0 to 180 degrees), which means it will always give you an angle from 0 to 180 degrees, suitable for triangles since triangle angles fall within this range. When using a calculator to find inverse cosine, ensure it is in the correct mode (degrees or radians) corresponding to the problem you are solving.

Triangle Geometry

Triangle geometry is the study of the properties and relations of triangles. It encompasses various concepts such as angles, sides, heights, bisectors, and medians. In any triangle, the sum of the interior angles is always 180 degrees. This fact is essential in conjunction with the Law of Cosines, as it allows us to determine the third angle after finding two of them.
Triangles can be classified based on their sides (scalene, isosceles, or equilateral) or angles (acute, right, or obtuse). Each classification has different properties and formulas associated with it. For instance, an equilateral triangle's equal sides mean knowing one side gives you information about all three. Understanding triangle geometry enables you to use the Law of Cosines and other trigonometric identities more effectively, making it a critical aspect of solving any triangle-related problems.