Problem 43 - Algebra 2

Question

In \(\Delta R S T, t=7 \mathrm{ft}\) and \(s=13 \mathrm{ft}\) . Find each value to the nearest tenth. Find \(m \angle S\) for \(r=14 \mathrm{ft}\)

Step-by-Step Solution

Answer

The value of \( m \angle S \) to the nearest tenth, when \( t = 7 \mathrm{ft}, s = 13 \mathrm{ft}, r = 14 \mathrm{ft} \) is approximately 34.1°.

Step 1: Using the Law of Cosines

The law of cosines is expressed as: \( c^2 = a^2 + b^2 - 2ab \cos C \). Since we need to solve for \( m \angle S \), in the triangle context, let \( r = a, t = b, s= c \) and \( \angle S = C \). So our law of cosines equation will be: \( s^2 = r^2 + t^2 - 2rt \cos S \).

Step 2: Setting up and Solving the Equation

We slot in the given values and solve for \( \cos S \), yielding: \( 13^2 = 14^2 + 7^2 - 2*14*7*\cos S \). On simplifying, we find \( \cos S = \frac{14^2 + 7^2 - 13^2}{2*14*7} \).

Step 3: Finding \( m \angle S \)

Taking inverse cosine to get the value of \( m \angle S \), we solve: \( S = \arccos(\frac{14^2 + 7^2 - 13^2}{2*14*7}) \). Therefore, \( m \angle S \) to the nearest tenth is computed by evaluating the expression inside the arccos function.

Key Concepts

Solving Triangles

Solving triangles involves finding unknown sides or angles within a triangle, using the given information and mathematical concepts. This is a fundamental aspect of trigonometry. In our example, we are tasked with finding the measure of an angle when the side lengths are given.
Key points to remember when solving triangles are:

• Identify what is given and what needs to be found. This helps in choosing the right formula or method.
• Apply the Law of Cosines if you know two sides and the included angle or three sides but no angles.

In this case, you have a triangle with sides measuring 7 ft, 13 ft, and 14 ft. Your challenge is to determine the angle opposite the side that measures 13 ft, using the Law of Cosines. This approach helps in systematically finding the unknown measure by relating it to the known sides.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It is especially useful in solving triangles using formulas like the Law of Cosines and the Law of Sines.
The Law of Cosines serves a similar purpose to the Pythagorean theorem but is used for any triangle, not just right triangles. It is expressed as: \[c^2 = a^2 + b^2 - 2ab \cos C\]This formula allows you to find an unknown side or angle when you have at least three pieces of information about a triangle.

• "c" is the side opposite the angle "C" you are solving for.
• "a" and "b" are the other two sides of the triangle.

By applying this, you convert the geometric information into a solvable equation. In our exercise, we substitute the respective side lengths into the Law of Cosines, making it possible to isolate and solve for the angle.

Inverse Cosine

To solve for an angle when you know the cosine, typically found via the Law of Cosines in trigonometry, you use the inverse cosine function (also known as arccos).
The inverse cosine function helps find the angle whose cosine is the given number. After rearranging the Law of Cosines formula to solve for \[\cos C = \frac{a^2 + b^2 - c^2}{2ab}\]we use the inverse cosine function to find the actual angle.Steps using inverse cosine:

• Compute the value from the equation \(\frac{a^2 + b^2 - c^2}{2ab}\).
• Apply the inverse cosine function to find the angle: \(C = \arccos(\cos C)\).

In the context of our exercise, after obtaining the cosine value, the inverse cosine gives us the measure of angle \(S\) to the nearest tenth, completing the problem.