Problem 10 - Algebra 2 and Trigonometry

Question

In \(\triangle D E F, \mathrm{m} \angle D=56, \mathrm{m} \angle E=44,\) and \(d=37.5 .\) Find \(e\) to the nearest tenth.

Step-by-Step Solution

Answer

The length of \(e\) is approximately 31.4.

Step 1: Understand the Angles

In any triangle, the sum of the interior angles is always 180 degrees. In \(\triangle DEF\), we are given that \(\mathrm{m}\angle D = 56\) and \(\mathrm{m}\angle E = 44\). We can use this information to find the measure of angle \(\angle F\).

Step 2: Calculate Measure of Angle F

Use the formula for the sum of angles in a triangle: \(\mathrm{m}\angle D + \mathrm{m}\angle E + \mathrm{m}\angle F = 180\). Substitute the known values: \(56 + 44 + \mathrm{m}\angle F = 180\). Simplify to find \(\mathrm{m}\angle F = 180 - 56 - 44 = 80\).

Step 3: Use the Law of Sines

The Law of Sines states that \(\frac{d}{\sin(D)} = \frac{e}{\sin(E)}\). We are given \(d = 37.5\), \(\mathrm{m}\angle D = 56\), and \(\mathrm{m}\angle E = 44\). We can use the Law of Sines to solve for \(e\).

Step 4: Calculate Sin of Angles

Compute \(\sin(56^\circ)\) and \(\sin(44^\circ)\) using a calculator, which yields \(\sin(56^\circ) \approx 0.829\) and \(\sin(44^\circ) \approx 0.695\).

Step 5: Solve for e

Substitute the known values into the Law of Sines equation: \(\frac{37.5}{0.829} = \frac{e}{0.695}\). Solve for \(e\) by multiplying both sides by \(0.695\) to get \(e = 37.5 \times \frac{0.695}{0.829}\). Simplify this to find \(e \approx 31.4\).

Step 6: Round to the Nearest Tenth

Based on the solution from Step 5, we concluded that \(e \approx 31.4\), which is already to the nearest tenth.

Key Concepts

Triangles

Triangles are simple but crucial shapes in geometry. They consist of three sides and three angles. The sum of these angles always adds up to 180 degrees, no matter what type of triangle it is.
A triangle is named by its vertices, for example, triangle DEF, indicating the corners of the triangle.
It's important to know about the different types of triangles based on their angles:

• **Acute triangle**: All angles are less than 90 degrees.
• **Right triangle**: One angle is exactly 90 degrees.
• **Obtuse triangle**: One angle is more than 90 degrees.

Each type has its own property and they are used in different applications like construction, art, and even computer graphics. Understanding how to work with triangles will sharpen your math skills and problem-solving capabilities.

Sum of Angles in a Triangle

The concept of the sum of angles in a triangle is fundamental in geometry. Every triangle, regardless of its type (whether it's acute, right, or obtuse), has interior angles that add up to a perfect 180 degrees.
This rule helps in solving various problems related to triangles. For instance, if you know two angles in a triangle, you can easily find the third angle by subtracting the sum of the known angles from 180.

• In the exercise, knowing angles D and E allowed us to find angle F by using the equation: \( \mathrm{m}\angle D + \mathrm{m}\angle E + \mathrm{m}\angle F = 180\).
• It's a straightforward calculation: \( 56 + 44 + \mathrm{m}\angle F = 180\).
• Solving gives \( \mathrm{m}\angle F = 80 \) degrees.

This principle is not only important theoretically but also practically, as it is foundational for understanding more complex mathematical concepts.

Trigonometric Functions

Trigonometric functions are mathematical tools used to relate angles and sides of triangles. In particular, they are useful for non-right triangles, thanks to the **Law of Sines**.
The Law of Sines states that:

• For any triangle with sides a, b, c, opposite angles A, B, C respectively, \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \).

This equation helps us find missing side lengths or angles. In the exercise, knowing \( \angle D = 56\) and \( \angle E = 44\), as well as side d of 37.5, allowed us to calculate side e using:

• Calculate \( \sin(56^\circ) \) and \( \sin(44^\circ) \).
• Use \( \frac{37.5}{0.829} = \frac{e}{0.695} \) to find \( e \).
• Solve for \(e\), yielding \( e \approx 31.4 \).

Understanding trigonometric functions and how to use them to solve problems in triangles is an essential part of learning geometry and will aid in more advanced topics like calculus.