Problem 41 - 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 \(m \angle T\) for \(r=11 \mathrm{ft}\)

Step-by-Step Solution

Answer

The measure of angle T in the triangle to the nearest tenth is calculated using the law of cosines and then rounded the answer to the nearest tenth.

Step 1: Identify the given values

In the triangle \( \Delta RST \), the given sides are \(s = 13\), \( t = 7\) and \(r = 11\). We are to find \(m \angle T\).

Step 2: Apply the Law of Cosines

The law of cosines states that for any given triangle with sides of lengths a, b and c and the angle \( \gamma \) opposite to side c, it can be expressed as follows: \( c^2 = a^2 + b^2 - 2ab\cos(\gamma) \). To find \( m \angle T\) we can express and rearrange this formula to \( m \angle T = \cos^{-1}\left(\frac{s^2 + t^2 - r^2}{2st}\right) \)

Step 3: Substitution and Calculation

Substitute the given side lengths into the law of cosines formula, \( T = \cos^{-1}\left(\frac{13^2 + 7^2 - 11^2}{2*13*7}\right)\). Solve the equation for T. Be sure to use inverse cosine to find the measure of angle T.

Step 4: Rounding And Final Result

After solving the equation, round the result to the nearest tenth as specified in the problem. This will give the measure of angle T in degrees.

Key Concepts

Triangle

A triangle is a basic geometric shape with three sides and three angles. It's important because understanding triangles is foundational in geometry. The most common type of triangle you'll encounter is one where all sides are different lengths, called a scalene triangle. The sides are usually labeled as 'a', 'b', and 'c', while the angles opposite to these sides are labeled as angles A, B, and C.
These triangles are used to understand shape properties and solve various real-world problems. Different types of triangles, like equilateral (all sides equal) and isosceles (two sides equal), have their specific properties too, but the principles of solving them often rely on the same basic rules. Understanding how to label and work with the sides and angles of a triangle is crucial for problem-solving.

Angle Calculation

Calculating the angle in a triangle requires knowledge of its sides and the use of trigonometric principles. One such formula is the Law of Cosines, which is incredibly useful when you know the lengths of all three sides of a triangle.
The Law of Cosines formula helps to calculate an unknown angle given three sides. It states:

• For a triangle with sides 'a', 'b', and 'c', with the angle opposite side 'c' being \( \gamma \), the formula is: \( c^2 = a^2 + b^2 - 2ab\cos(\gamma) \).
• This equation can be rearranged to solve for \( \gamma \), which becomes \( \gamma = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right) \).

When solving for an angle, remember to use the appropriate inverse trigonometric function. This function helps convert a ratio back into a measurable angle. Practice calculating angles using the triangle's dimensions to fully understand how to manipulate and use these equations.

Inverse Cosine

The inverse cosine function, denoted as \( \cos^{-1} \), is crucial for finding angle measures from known cosine values. When you have calculated a fraction using the Law of Cosines' formula, the inverse cosine helps convert that value into degrees, which are more understandable and practical for depicting angles.
Using inverse cosine is straightforward:

• First, calculate the value needed from the equation.
• Then, input this value into the inverse cosine function. \( \cos^{-1}(x) \) returns an angle \( \theta \) such that \( \cos(\theta) = x \).

This process allows you to solve for angles when using the Law of Cosines. It's an essential skill to master as it translates the calculated number, often between -1 and 1, into a useful angle measure, typically within the range of 0° to 180° for triangles. Understanding how to use the inverse cosine shifts your focus from side lengths to the actual geometry's angular properties.