Problem 2 - Precalculus : Building Concepts and Connections

Question

Use the Law of Cosines to find the umknoten side of each of the given triangles. The standard notation for labeling of triangles is used. Round anstoers to four decimal places. $$b=8, c=4, A=75^{\circ}$$

Step-by-Step Solution

Answer

The value of side a can be found using the Law of Cosines and rounding the result to four decimal places.

Step 1: Ewrite the required Law of Cosines

From the given data, it is necessary to use the Law of Cosines relating sides a, b, c and angle A. Thus the law becomes \(a^2 = b^2 + c^2 -2bc \cdot cos(A)\).

Step 2: Substitute the given values into the formula

Substitute \(b=8, c=4, A=75^\circ\) into the formula. Thus the formula becomes \(a^2 = 8^2 + 4^2 -2 \cdot 8 \cdot 4 \cdot cos(75^\circ)\).

Step 3: Calculate \(a^2\)

Compute the right side to find the value of \(a^2\). Remember to convert angle A to radians when computing it with cosine, as most calculators work with radians.

Step 4: Find the side a

After finding \(a^2\), calculate the value of side a by taking the square root of the result. Round the answer to four decimal places, if needed.

Key Concepts

Triangle Calculations

Triangles are the basic building blocks in geometry, often met in various mathematical problems. The main goal in triangle calculations is to determine unknown sides or angles. Many formulas and laws exist to help with this, one being the Law of Cosines.
This particular law is essential when you are given two sides and the included angle or need to find an unknown side when you know two angles and a side. In our exercise example, we apply the Law of Cosines:

• Given sides are typically labeled as sides b, c, and angle A across from the unknown side a.
• The formula in its full form is: \(a^2 = b^2 + c^2 - 2bc \cdot \cos(A)\).

These calculations require a step-by-step approach where each term is substituted, calculated, and evaluated to finally arrive at the needed side length.

Trigonometry

Trigonometry is a branch of mathematics dealing with the relationships between the angles and sides of triangles. It is a crucial aspect when solving problems involving triangles, especially right-angled ones, through the algorithms like sine, cosine, and tangent.
The Law of Cosines is an extension of the trigonometric principles that allows for calculations beyond right-angled triangles. In this law, using cosine function helps link angles and sides in any triangle configuration. Calculating the cosine of an angle, as seen in our exercise, involves understanding the role of cosine in defining the relationship between side lengths and angles across from them.
Trigonometric identities and laws are fundamental for precise calculations, especially when working in physics and engineering contexts where accuracy is paramount.

Angle Conversion

An essential skill often needed in trigonometry is converting angles between degrees and radians. Most scientific calculators default to using radians in functions like cosine, so ensuring consistency in angle units is crucial to get accurate results.
A simple conversion can be done using the formula:

• \(\text{Radians} = \text{Degrees} \times \left(\frac{\pi}{180}\right)\)

In the given exercise, we start with an angle of 75 degrees. If you're calculating the cosine of this angle, it's crucial to first convert it into radians:

• \(75^{\circ} = 75 \times \left(\frac{\pi}{180}\right) = \text{1.309 radians}\)

Using radians ensures compatibility with most calculation tools and reduces errors in otherwise precise trigonometric calculations.