Upload your work... Draw the triangle in the correct quadrant with ALL appropriate labels. This includes finding the MISSING SIDE. If $$ \sin heta=1 / 3 $$ and $$ heta $$ is in Quad ॥

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Upload your work... Draw the triangle in the correct quadrant with ALL appropriate labels. This includes finding the MISSING SIDE. If $$ \sin 	heta=1 / 3 $$ and $$ 	heta $$ is in Quad ॥

UpStudy AI · Accepted Answer

Certainly! Let's work through the problem step by step to draw the appropriate right triangle, label all parts, determine the missing side, and indicate the correct quadrant for angle $$\theta$$.

### Given:
- $$\sin \theta = \frac{1}{3}$$
- $$\theta$$ is in **Quadrant II**.

### Understanding the Quadrant:
In **Quadrant II**:
- **Sine** values are **positive**.
- **Cosine** and **Tangent** values are **negative**.

### Step 1: Draw the Coordinate Plane and Identify Quadrant II

```
|
|        Quadrant II
|        
|        
|-------------------------
|
|
```

### Step 2: Draw the Right Triangle in Quadrant II

1. **Angle $$\theta$$** is positioned such that its vertex is at the origin, and its terminal side extends into Quadrant II.
2. Draw a perpendicular from the terminal side to the x-axis to form a right triangle.

```
          |
          |\
          | \
        1 |  \ 3
          |   \
          |____\
           -x
```

- **Opposite Side (Vertical side)**: 1 (since $$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{1}{3}$$)
- **Hypotenuse**: 3
- **Adjacent Side (Horizontal side)**: To be determined (denoted as $$-x$$ because in Quadrant II, the adjacent side is negative)

### Step 3: Determine the Missing Side Using the Pythagorean Theorem

The Pythagorean Theorem states:
$$
\text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2
$$

Plugging in the known values:
$$
3^2 = 1^2 + (-x)^2 \
9 = 1 + x^2 \
x^2 = 9 - 1 \
x^2 = 8 \
x = \sqrt{8} = 2\sqrt{2}
$$

Since the adjacent side is in Quadrant II, it's negative:
$$
\text{Adjacent} = -2\sqrt{2}
$$

### Step 4: Label All Parts of the Triangle

```
          |
          |\
          | \
        1 |  \ 3
          |   \
          |____\
        -2√2  x-axis
```

- **Angle $$\theta$$**: Located at the origin.
- **Opposite Side**: 1 (vertical side)
- **Adjacent Side**: $$-2\sqrt{2}$$ (horizontal side, negative in Quadrant II)
- **Hypotenuse**: 3

### Final Diagram Description

- **Origin (0,0)**: Vertex of angle $$\theta$$.
- **Terminal Side of $$\theta$$**: Extends into Quadrant II.
- **Opposite Side**: Length of 1, along the y-axis.
- **Adjacent Side**: Length of $$2\sqrt{2}$$, extending into the negative direction of the x-axis.
- **Hypotenuse**: Connecting the origin to the point $$(-2\sqrt{2}, 1)$$, length 3.

### Summary

- **$$\sin \theta = \frac{1}{3}$$** implies:
  - **Opposite Side**: 1
  - **Hypotenuse**: 3
- **Using the Pythagorean Theorem**, the **Adjacent Side** is found to be $$-2\sqrt{2}$$ (negative in Quadrant II).
- All parts are labeled appropriately in the right triangle positioned in **Quadrant II**.

If you need a visual diagram, here's a textual representation:

```
Quadrant II

|
       |\
       | \
     1 |  \ 3
       |   \
       |____\
     -2√2  x-axis
```

Feel free to recreate this diagram on graph paper or using a drawing tool for a clearer visual understanding!
To solve the problem: 1. **Given**: - $$\sin \theta = \frac{1}{3}$$ - $$\theta$$ is in Quadrant II. 2. **Draw a right triangle** in Quadrant II with: - **Opposite side** = 1 - **Hypotenuse** = 3 - **Adjacent side** = $$-2\sqrt{2}$$ (negative in Quadrant II) 3. **Label all parts** of the triangle accordingly. 4. **Final Diagram**: - Angle $$\theta$$ at the origin. - Opposite side along the y-axis: length 1. - Adjacent side along the negative x-axis: length $$2\sqrt{2}$$. - Hypotenuse connecting the origin to $$(-2\sqrt{2}, 1)$$, length 3. This setup ensures all parts are correctly labeled and the triangle is accurately positioned in Quadrant II.

Upload your work…
Draw the triangle in the correct quadrant with ALL appropriate labels.
This includes finding the MISSING SIDE.
If and is in Quad ॥

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Upload your work… Draw the triangle in the correct quadrant with ALL appropriate labels. This includes finding the MISSING SIDE. If and is in Quad ॥