aing Classkick
L52-Rt Angle Ting (Th)
Question 11 of 25 (1 point) । Question Aftempt 1 of 5
First use the Pythagorean theorem to find the exact length of the missing side of the right triangle. Then find the exact values of the six trigonometric functions
for the angle opposite the shortest side.
Leg , Hypotenuse
Part 1 of 2
Hypotenuse , Leg , Leg .

Ask by Ayala Bowers. in the United States

Jan 22,2025

Question

aing Classkick
L52-Rt Angle Ting (Th)
Question 11 of 25 (1 point) । Question Aftempt 1 of 5
First use the Pythagorean theorem to find the exact length of the missing side of the right triangle. Then find the exact values of the six trigonometric functions
for the angle opposite the shortest side.
Leg , Hypotenuse
Part 1 of 2
Hypotenuse , Leg , Leg .

Ask by Ayala Bowers. in the United States

Jan 22,2025

UpStudy AI · Accepted Answer

Sure, let's work through the problem step by step.

### **Given:**
- **One leg (adjacent to angle $$ \theta $$)**: $$ 4\sqrt{23} $$ meters
- **Hypotenuse**: $$ 6\sqrt{13} $$ meters
- **Missing leg (opposite to angle $$ \theta $$)**: $$ 10 $$ meters

### **1. Pythagorean Theorem Verification**
First, let's confirm the missing leg using the Pythagorean theorem:

$$
\text{Hypotenuse}^2 = (\text{Adjacent side})^2 + (\text{Opposite side})^2
$$

$$
(6\sqrt{13})^2 = (4\sqrt{23})^2 + (10)^2
$$

$$
36 \times 13 = 16 \times 23 + 100
$$

$$
468 = 368 + 100
$$

$$
468 = 468 \quad \text{✓}
$$

The missing leg is indeed $$ 10 $$ meters.

### **2. Trigonometric Functions for Angle $$ \theta $$**
**Angle $$ \theta $$** is opposite the shortest side ($$ 10 $$ meters). We'll find all six trigonometric functions based on the sides of the triangle.

#### **a. Sine ($$ \sin \theta $$)**
$$
\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{10}{6\sqrt{13}} = \frac{5}{3\sqrt{13}} = \frac{5\sqrt{13}}{39}
$$

#### **b. Cosine ($$ \cos \theta $$)**
$$
\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4\sqrt{23}}{6\sqrt{13}} = \frac{2\sqrt{23}}{3\sqrt{13}} = \frac{2\sqrt{299}}{39}
$$

#### **c. Tangent ($$ \tan \theta $$)**
$$
\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{10}{4\sqrt{23}} = \frac{5}{2\sqrt{23}} = \frac{5\sqrt{23}}{46}
$$

#### **d. Cosecant ($$ \csc \theta $$)**
$$
\csc \theta = \frac{1}{\sin \theta} = \frac{39}{5\sqrt{13}} = \frac{39\sqrt{13}}{65} = \frac{3\sqrt{13}}{5}
$$

#### **e. Secant ($$ \sec \theta $$)**
$$
\sec \theta = \frac{1}{\cos \theta} = \frac{39}{2\sqrt{299}} = \frac{39\sqrt{299}}{598} = \frac{3\sqrt{299}}{46}
$$

#### **f. Cotangent ($$ \cot \theta $$)**
$$
\cot \theta = \frac{1}{\tan \theta} = \frac{2\sqrt{23}}{5}
$$

### **Summary of Trigonometric Functions:**

- **$$\sin \theta = \frac{5\sqrt{13}}{39}$$**
- **$$\cos \theta = \frac{2\sqrt{299}}{39}$$**
- **$$\tan \theta = \frac{5\sqrt{23}}{46}$$**
- **$$\csc \theta = \frac{3\sqrt{13}}{5}$$**
- **$$\sec \theta = \frac{3\sqrt{299}}{46}$$**
- **$$\cot \theta = \frac{2\sqrt{23}}{5}$$**

These are the exact values of the six trigonometric functions for angle $$ \theta $$ in the given right triangle.
The exact trigonometric functions for angle $$ \theta $$ are: - **$$\sin \theta = \frac{5\sqrt{13}}{39}$$** - **$$\cos \theta = \frac{2\sqrt{299}}{39}$$** - **$$\tan \theta = \frac{5\sqrt{23}}{46}$$** - **$$\csc \theta = \frac{3\sqrt{13}}{5}$$** - **$$\sec \theta = \frac{3\sqrt{299}}{46}$$** - **$$\cot \theta = \frac{2\sqrt{23}}{5}$$**

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