**Problem 3: Solving for Unknown Side Lengths and Angles in Right ∆DEF****Given:** - Right triangle ∆DEF - $ DE = 2.1 $ - $ EF = 5.3 $ **Objective:** Use a calculator and inverse trigonometric ratios to find the unknown side lengths and angle measures. Round lengths to the nearest hundredth and angle measures to the nearest degree.**Diagram Explanation:**There is a right triangle labeled ∆DEF where:- $ DE $ is one of the legs of the triangle with a length of 2.1 units.- $ EF $ is the other leg of the triangle with a length of 5.3 units.- $ DF $ is the hypotenuse, which we need to find.### Steps:1. **Find the Hypotenuse $ DF $:** Using the Pythagorean Theorem: \[ DF = \sqrt{DE^2 + EF^2} \] \[ DF = \sqrt{(2.1)^2 + (5.3)^2} \] \[ DF = \sqrt{4.41 + 28.09} \] \[ DF = \sqrt{32.50} \] \[ DF \approx 5.70 \]2. **Find the Angle $ ∠DEF $:** Using the tangent function: \[ \tan(∠DEF) = \frac{DE}{EF} = \frac{2.1}{5.3} \] Finding the angle using the arctangent function: \[ ∠DEF = \tan^{-1}\left(\frac{2.1}{5.3}\right) \] \[ ∠DEF \approx \tan^{-1}(0.3962) \approx 22^\circ \]3. **Find the Angle $ ∠DFE $:** The sum of the angles in any triangle is 180 degrees. In a right triangle, we already have a 90-degree angle at $ ∠DEF $. Thus, \[ ∠DFE = 90^\circ - ∠DEF \] \[ ∠DFE = 90^\

The given right triangle has DE=2.1 , EF=5.3 and angle E is 90 degree.

3. Given right ADEF, DE = 2.1 and EF = 5.3. Use a calculator and inverse trigonometric ratios to find the unknown side lengths and angle measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 2.1 5.3

3. Given right ADEF, DE = 2.1 and EF = 5.3. Use a calculator and inverse trigonometric ratios to find the unknown side lengths and angle measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 2.1 5.3

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter3: Additional Topics In Trigonometry

Section: Chapter Questions

Problem 40CT: To determine the angle of elevation of a star in the sky, you align the star and the top of the...

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Question

**Problem 3: Solving for Unknown Side Lengths and Angles in Right ∆DEF**

**Given:**
- Right triangle ∆DEF
- $ DE = 2.1 $
- $ EF = 5.3 $

**Objective:**
Use a calculator and inverse trigonometric ratios to find the unknown side lengths and angle measures. Round lengths to the nearest hundredth and angle measures to the nearest degree.

**Diagram Explanation:**
There is a right triangle labeled ∆DEF where:
- $ DE $ is one of the legs of the triangle with a length of 2.1 units.
- $ EF $ is the other leg of the triangle with a length of 5.3 units.
- $ DF $ is the hypotenuse, which we need to find.

### Steps:
1. **Find the Hypotenuse $ DF $:**

Using the Pythagorean Theorem:
\[
DF = \sqrt{DE^2 + EF^2}
\]
\[
DF = \sqrt{(2.1)^2 + (5.3)^2}
\]
\[
DF = \sqrt{4.41 + 28.09}
\]
\[
DF = \sqrt{32.50}
\]
\[
DF \approx 5.70
\]

2. **Find the Angle $ ∠DEF $:**

Using the tangent function:
\[
\tan(∠DEF) = \frac{DE}{EF} = \frac{2.1}{5.3}
\]

Finding the angle using the arctangent function:
\[
∠DEF = \tan^{-1}\left(\frac{2.1}{5.3}\right)
\]
\[
∠DEF \approx \tan^{-1}(0.3962) \approx 22^\circ
\]

3. **Find the Angle $ ∠DFE $:**

The sum of the angles in any triangle is 180 degrees. In a right triangle, we already have a 90-degree angle at $ ∠DEF $. Thus,
\[
∠DFE = 90^\circ - ∠DEF
\]
\[
∠DFE = 90^\

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