Type or upload your work and solution.
Your work should include right triangles in the appropriate quadrants with
appropriate labels
If and is in Quad III and and is in Quad II.
Find the following:
a)
b)
c) is in Quadrant

Ask by Bernard Nunez. in the United States

Jan 27,2025

Question

Type or upload your work and solution.
Your work should include right triangles in the appropriate quadrants with
appropriate labels
If and is in Quad III and and is in Quad II.
Find the following:
a)
b)
c) is in Quadrant

Ask by Bernard Nunez. in the United States

Jan 27,2025

UpStudy AI · Accepted Answer

Sure! Let's work through each part of the problem step by step. We'll use right triangles in the appropriate quadrants and apply trigonometric identities to find the solutions.

---

## Given:
1. $$ \cos \alpha = -\frac{3}{5} $$ and $$ \alpha $$ is in **Quadrant III**.
2. $$ \tan \beta = -\frac{5}{12} $$ and $$ \beta $$ is in **Quadrant II**.

## Objectives:
a) Find $$ \cos (\alpha + \beta) $$

b) Find $$ \sin (\alpha + \beta) $$

c) Determine the quadrant in which $$ (\alpha + \beta) $$ lies.

---

## Part a) $$ \cos (\alpha + \beta) $$

### Step 1: Recall the Cosine Addition Formula
$$
\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta
$$

### Step 2: Find $$ \sin \alpha $$ and $$ \cos \beta $$, $$ \sin \beta $$

#### For $$ \alpha $$ in Quadrant III:
- $$ \cos \alpha = -\frac{3}{5} $$
- In Quadrant III, both sine and cosine are negative.

**Constructing the Right Triangle for $$ \alpha $$:**
- Adjacent side: 3 (negative)
- Hypotenuse: 5
- Opposite side: $$ y $$ (to be found)

Using Pythagoras:
$$
y = -\sqrt{5^2 - 3^2} = -\sqrt{25 - 9} = -\sqrt{16} = -4
$$
$$
\sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{-4}{5}
$$

#### For $$ \beta $$ in Quadrant II:
- $$ \tan \beta = -\frac{5}{12} $$ 
- In Quadrant II, sine is positive and cosine is negative.

**Constructing the Right Triangle for $$ \beta $$:**
- Opposite side: 5 (since tangent is opposite/adjacent, and tangent is negative in Quadrant II, adjacent side is negative)
- Adjacent side: -12
- Hypotenuse: $$ r $$ (to be found)

Using Pythagoras:
$$
r = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
$$
$$
\cos \beta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{-12}{13}
$$
$$
\sin \beta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{5}{13}
$$

### Step 3: Apply the Cosine Addition Formula
$$
\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta
$$
$$
= \left(-\frac{3}{5}\right) \left(-\frac{12}{13}\right) - \left(\frac{-4}{5}\right) \left(\frac{5}{13}\right)
$$
$$
= \frac{36}{65} - \left(-\frac{20}{65}\right)
$$
$$
= \frac{36}{65} + \frac{20}{65} = \frac{56}{65}
$$

**Answer for Part a):**
$$
\cos (\alpha + \beta) = \frac{56}{65}
$$

---

## Part b) $$ \sin (\alpha + \beta) $$

### Step 1: Recall the Sine Addition Formula
$$
\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
$$

### Step 2: Use the Previously Found Values
$$
\sin (\alpha + \beta) = \left(\frac{-4}{5}\right) \left(-\frac{12}{13}\right) + \left(-\frac{3}{5}\right) \left(\frac{5}{13}\right)
$$
$$
= \frac{48}{65} + \left(-\frac{15}{65}\right)
$$
$$
= \frac{48}{65} - \frac{15}{65} = \frac{33}{65}
$$

**Answer for Part b):**
$$
\sin (\alpha + \beta) = \frac{33}{65}
$$

---

## Part c) Determine the Quadrant of $$ (\alpha + \beta) $$

To determine the quadrant where $$ (\alpha + \beta) $$ lies, we consider the signs of $$ \cos (\alpha + \beta) $$ and $$ \sin (\alpha + \beta) $$.

From Parts a) and b):
$$
\cos (\alpha + \beta) = \frac{56}{65} > 0
$$
$$
\sin (\alpha + \beta) = \frac{33}{65} > 0
$$

### Step 1: Analyze the Signs
- Both sine and cosine of $$ (\alpha + \beta) $$ are positive.

### Step 2: Recall Quadrant Sign Rules
- **Quadrant I:** Both sine and cosine are positive.
- **Quadrant II:** Sine is positive, cosine is negative.
- **Quadrant III:** Both sine and cosine are negative.
- **Quadrant IV:** Sine is negative, cosine is positive.

### Conclusion:
Since both $$ \sin (\alpha + \beta) $$ and $$ \cos (\alpha + \beta) $$ are positive, $$ (\alpha + \beta) $$ lies in **Quadrant I**.

**Answer for Part c):**
$$
(\alpha + \beta) \text{ is in Quadrant I.}
$$

---

## Summary of Answers:
a) $$ \cos (\alpha + \beta) = \frac{56}{65} $$

b) $$ \sin (\alpha + \beta) = \frac{33}{65} $$

c) $$ (\alpha + \beta) $$ is in **Quadrant I**

---

### Visual Representation (Right Triangles)

While I can't provide actual drawings here, I'll describe the right triangles for both angles in their respective quadrants.

#### Triangle for $$ \alpha $$ in Quadrant III:
- **Adjacent side (x):** -3
- **Opposite side (y):** -4
- **Hypotenuse (r):** 5

#### Triangle for $$ \beta $$ in Quadrant II:
- **Adjacent side (x):** -12
- **Opposite side (y):** 5
- **Hypotenuse (r):** 13

These triangles help in determining the sine and cosine values based on the given trigonometric ratios and their signs in the respective quadrants.

---

I hope this clarifies the problem and its solution! Let me know if you have any further questions.
$$ \cos (\alpha + \beta) = \frac{56}{65} $$, $$ \sin (\alpha + \beta) = \frac{33}{65} $$, and $$ (\alpha + \beta) $$ is in Quadrant I.

Type or upload your work and solution.
Your work should include right triangles in the appropriate quadrants with
appropriate labels
If and is in Quad III and and is in Quad II.
Find the following:
a)
b)
c) is in Quadrant

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Type or upload your work and solution. Your work should include right triangles in the appropriate quadrants with appropriate labels If and is in Quad III and and is in Quad II. Find the following: a) b) c) is in Quadrant

Solución de inteligencia artificial de Upstudy

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preguntas relacionadas

Latest Trigonometry Questions

Type or upload your work and solution.
Your work should include right triangles in the appropriate quadrants with
appropriate labels
If and is in Quad III and and is in Quad II.
Find the following:
a)
b)
c) is in Quadrant