Problem 9 - Algebra 2 and Trigonometry

Question

In \(\triangle D E F, \mathrm{m} \angle D=47, \mathrm{m} \angle E=84,\) and \(d=17.3 .\) Find \(e\) to the nearest tenth.

Step-by-Step Solution

Answer

The length of side \( e \) is approximately 23.5.

Step 1: Determine the Measure of Angle F

In any triangle, the sum of the interior angles is always 180 degrees. Use this property to find the measure of angle \( F \) in \( \triangle DEF \). We have \( \mathrm{m} \angle D = 47^\circ \) and \( \mathrm{m} \angle E = 84^\circ \), therefore, \( \mathrm{m} \angle F = 180^\circ - 47^\circ - 84^\circ = 49^\circ \).

Step 2: Use the Law of Sines

The Law of Sines states that \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). For \( \triangle DEF \), we use the known side \( d = 17.3 \) opposite to angle \( D \) and angle \( E \) to find side \( e \). Thus, it becomes: \( \frac{d}{\sin D} = \frac{e}{\sin E} \).

Step 3: Plug in Known Values and Solve for e

Substitute the known values into the Law of Sines: \( \frac{17.3}{\sin 47^\circ} = \frac{e}{\sin 84^\circ} \). Calculate \( \sin 47^\circ \) and \( \sin 84^\circ \), then solve for \( e \): \[ 17.3 \times \frac{\sin 84^\circ}{\sin 47^\circ} = e \]. After calculation, \( e \approx 23.5 \).

Step 4: Round the Answer

Finally, round the calculated value of \( e \) to the nearest tenth. We already have \( e \approx 23.5 \). Therefore, the length of side \( e \) is 23.5.

Key Concepts

Understanding Triangle Interior Angles

When we talk about triangles, one important property to remember is that all the interior angles add up to 180 degrees. This fundamental rule helps us find missing angles whenever we know at least two of the angles in a triangle.
For example, consider a triangle where two angles are known. To find the third angle, like in the problem with triangle DEF, you subtract the sum of the known angles from 180.

• The triangle's total is 180 degrees. If two angles are 47° and 84°:
• Add: 47° + 84° = 131°.
• Subtract from 180°: 180° - 131° = 49°.

This is how we find that angle F measures 49 degrees. Knowing the interior angles is crucial, especially when solving more complex trigonometry problems.

How to Calculate Angles Effectively

When calculating angles in a triangle, systematic steps can make your work simpler. Since the triangle's angles must total 180 degrees, you can use subtraction once the other angles are given.

• Begin with the sum of all angles, 180°.
• Subtract each known angle in steps.
• For triangle DEF: Start with 180°, subtract 47°, and then subtract 84°.

This practical approach ensures you've accounted for all angles. It's this simple process that informs angle calculations in geometry and beyond. Remember, having a firm grasp on these basics helps in solving trigonometric problems effectively.

Trigonometry Problem Solving with Law of Sines

Trigonometry often involves solving for unknown sides or angles in triangles using rules like the Law of Sines. This law relates the sides of a triangle to the sines of its angles, making it helpful in many geometric problems.

• The Law of Sines formula: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \) helps find missing sides.
• In triangle DEF, with side d = 17.3 and angles measured, we set: \( \frac{17.3}{\sin 47°} = \frac{e}{\sin 84°} \).
• Calculate \( \sin 47° \) and \( \sin 84° \), and solve for \( e \) by cross multiplying and simplifying.

Knowing how to use such laws is essential for trigonometry, providing a systematic way to approach many geometry and physics problems. Success in these problem-solving scenarios requires a careful, step-by-step application of concepts like the Law of Sines.