Problem 11 - Algebra 2

Question

In \(\triangle A B C, a=20 \mathrm{m}, b=14 \mathrm{m},\) and \(c=16 \mathrm{m} .\) Find \(m \angle A\)

Step-by-Step Solution

Answer

Using the given side lengths and the Law of Cosines, the measure of angle A in the triangle can be found to be approximately 46.19 degrees.

Step 1: Understanding the problem

You are given a triangle ABC with sides a=20m, b=14m and c=16m. You are asked to solve for the measure of angle A.

Step 2: Apply the Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle with the cosine of one of its angles. For this problem, we want to find angle A, given sides b, c, and a. Re-arranged to solve for \(\cos(A)\), the Law of Cosines is \(\cos(A) = \frac{b^{2}+c^{2}-a^{2}}{2bc}\). Substitute the given values into the formula.

Step 3: Simplify the expression

Calculate \(\frac{b^{2}+c^{2}-a^{2}}{2bc}\) using the given lengths b=14, c=16 and a=20. This will give you the cosine of angle A.

Step 4: Find the Angle

The arccosine (or inverse cosine) function can be used to find the angle from the cosine value. Use a calculator to find \( \arccos \) of the previously calculated value.

Key Concepts

Triangle Properties

When studying triangles, it's important to remember a few key properties. Triangles are closed shapes with three sides, and their angles always add up to 180 degrees. In a triangle, the side lengths and angles are connected through various theorems and formulas. One such theorem is the Law of Cosines, which is especially useful when you know all the side lengths and need to find an angle.

• Each angle in a triangle is opposite a specific side, and the length of that side can influence the angle's size.
• Triangles can be classified based on their angles: acute (all angles less than 90 degrees), right (one 90-degree angle), or obtuse (one angle more than 90 degrees).
• The sides of a triangle satisfy the triangle inequality: the sum of the lengths of any two sides must be greater than the length of the remaining side.

Understanding these properties is fundamental to solving problems involving triangles and their relationships.

Trigonometric Functions

Trigonometric functions are essential tools in geometry, allowing us to relate angles and sides in triangles. The primary trigonometric functions include sine, cosine, and tangent, often abbreviated as sin, cos, and tan.
For any angle in a triangle, these functions can express the ratio of two side lengths. In the context of the Law of Cosines used to solve for an angle, the cosine function becomes a bridge:

• Cosine of an angle refers to the ratio of the length of the adjacent side to the hypotenuse in a right triangle.
• In non-right triangles, the cosine function still serves as a key tool, part of the Law of Cosines formula.
• Understanding the trigonometric functions helps in solving for unknown angles or sides, especially when extending from right triangles to any arbitrary triangle.

Using these functions allows seamless navigation from the specific side lengths to the corresponding angles, given their inherent mathematical relationships.

Inverse Cosine

The inverse cosine, often denoted as \(\arccos\), is a function used to determine an angle when the cosine of that angle is known. This function is pivotal in finding angle measures from side lengths in a triangle.
The Law of Cosines first gives the cosine of an angle, but to find the actual angle, you must "reverse" the cosine process with \(\arccos\):

• The inverse cosine function takes a cosine value (ranging from -1 to 1) and returns the angle that corresponds to it, which lies between 0 and 180 degrees in the context of triangles.
• Using a calculator, the \( ext{arccos}\) function helps convert this ratio back into an angle, completing the solution to the problem.
• It's important to ensure that the calculated angle makes sense within the context of a triangle, checking against the triangle properties.

By using inverse cosine, you bridge the calculation from the spatial ratio to an actual angular measure, completing the necessary steps in trigonometric problem-solving.