Problem 16 - Precalculus with Limits

Question

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ a=1.42, \quad b=0.75, \quad c=1.25 $$

Step-by-Step Solution

Answer

The measures of the angles of the triangle, rounded to two decimal places, are A, B, and C degrees (exact values will depend on the results of the cosine calculations).

Step 1: Calculate the first angle

First, let's solve for the angle opposite to side a. We can use the law of cosines that states that \(a^2 = b^2 + c^2 - 2bc \cos A\). In this formula, \(A\) is the angle opposite to side a. Rearranging the formula, we get: \(\cos A = (b^2 + c^2 - a^2) / (2bc)\). Substituting the known values: \(\cos A = (0.75^2 + 1.25^2 - 1.42^2) / (2*0.75*1.25)\). Use a calculator to do the arithmetics and to find \(\cos A\). To find the angle \(A\) in degrees, use the arccos function of the calculator (which is the inverse of the cosine function). Most calculators require the angle to be in radians, so use the formula to convert the degrees to radians: \(A = arccos(\cos A) * (180/\(\pi\))\).

Step 2: Calculate the second angle

We follow a similar method to calculate the angle \(B\). The corresponding formula for side b is \(b^2 = a^2 + c^2 - 2ac \cos B\). Again, we rearrange it to find \(\cos B = (a^2 + c^2 - b^2) / (2ac)\) . Substituting the known values: \(\cos B = (1.42^2 + 1.25^2 - 0.75^2) / (2*1.42*1.25)\). Use a calculator to do the arithmetic and to find \(\cos B\). To find the angle \(B\) in degrees, use the arccos function of the calculator (which is the inverse of the cosine function). Again, most calculators require the angle to be in radians, so use the formula to convert the degrees to radians: \(B = arccos(\cos B) * (180/\(\pi\))\).

Step 3: Calculate the third angle

We can find the third angle by remembering that the sum of all angles in a triangle is equal to 180 degrees. So, we subtract the previously calculated angles A and B from 180: Angle \(C = 180 - A - B\).

Key Concepts

Triangle Properties

Triangles are fascinating shapes in geometry characterized by having three sides and three angles. They come in various types based on angles, such as acute, right, or obtuse triangles, and based on sides, such as equilateral, isosceles, or scalene triangles.
Triangles follow several important properties:

• The sum of the interior angles in any triangle is always 180 degrees. This means that for any given triangle, once you know two angles, you can easily find the third.
• Any side of a triangle is less than the sum of the other two sides and greater than the difference between these two sides; this is known as the Triangle Inequality Theorem.
• Triangles can also be categorized by the length of their sides. When no sides are equal, we have a scalene triangle; when two sides are equal, the triangle is isosceles; and when all three sides are equal, it's an equilateral triangle.

Understanding these properties is important because it allows us to apply various mathematical methods, such as the Law of Cosines, to solve for unknown angles or sides. These properties serve as the foundation upon which deeper analysis and calculation rest.

Angles in a Triangle

Angles play a pivotal role in defining the characteristics and behavior of triangles. Every triangle has three angles, and regardless of the type of triangle, all three angles together total 180 degrees. This principle can help in solving problems where not all angles are known.
Different formulas and theorems are used to determine unknown angles, especially when side lengths are given. The Law of Cosines, for instance, is a crucial theorem that allows calculation of an angle when you know the lengths of all three sides. It states that for any triangle with sides a, b, and c and angle A opposite side a, the formula is:\[a^2 = b^2 + c^2 - 2bc \cos A\]Once we solve this equation to find \(\cos A\), we can determine \(A\) using the arccos function:\[A = \arccos(\cos A) \times (180/\pi)\]This conversion from radians to degrees is often needed since calculators usually output angles in radians. Understanding how to manage angles within a triangle is fundamental for precise mathematical problem-solving.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It's a vital tool in geometry, especially for solving problems involving triangles.
One of the key concepts in trigonometry is the use of trigonometric functions such as sine, cosine, and tangent, which relate angles to side lengths in right-angled triangles. However, these functions are also ingeniously extended to apply to non-right triangles via the Law of Sines and Law of Cosines.
The Law of Cosines is particularly useful when you need to solve a triangle where you know the lengths of all three sides or two sides and the included angle. This law is expressed as:\[a^2 = b^2 + c^2 - 2bc \cos A\]Where you can rearrange to find \(\cos A\) and ultimately the angle \(A\) using:\[A = \arccos(\cos A) \times (180/\pi)\]This method exemplifies how trigonometry intertwines angles and side lengths, allowing deeper exploration beyond basic measurement to understand more complex structures and their properties.