If $$ \cos \alpha=0.487 $$ and $$ \cos \beta=0.658 $$ with both angles' terminal rays in Quadrant-1, find the values of (a) $$ \sin (\alpha+\beta)=\square $$ (b) $$ \sin (\beta-\alpha)=\square $$ Your answers should be accurate to 4 decimal places. Submit Question

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UpStudy AI · Accepted Answer

To solve the given problem, we'll use the following trigonometric identities:

1. **Sum Identity for Sine:**
   $$
   \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta
   $$
   
2. **Difference Identity for Sine:**
   $$
   \sin(\beta - \alpha) = \sin\beta \cos\alpha - \cos\beta \sin\alpha
   $$

Given:
$$
\cos\alpha = 0.487 \quad \text{and} \quad \cos\beta = 0.658
$$
Both angles $$\alpha$$ and $$\beta$$ are in Quadrant I, where sine values are positive.

### Step 1: Find $$\sin\alpha$$ and $$\sin\beta$$

Using the Pythagorean identity:
$$
\sin^2\theta + \cos^2\theta = 1 \implies \sin\theta = \sqrt{1 - \cos^2\theta}
$$

- **For $$\alpha$$:**
  $$
  \sin\alpha = \sqrt{1 - (0.487)^2} = \sqrt{1 - 0.237169} = \sqrt{0.762831} \approx 0.8734
  $$
  
- **For $$\beta$$:**
  $$
  \sin\beta = \sqrt{1 - (0.658)^2} = \sqrt{1 - 0.432964} = \sqrt{0.567036} \approx 0.7530
  $$

### Step 2: Calculate $$\sin(\alpha + \beta)$$

Using the sum identity:
$$
\sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta
$$
$$
\sin(\alpha + \beta) = (0.8734)(0.658) + (0.487)(0.7530) \approx 0.5747 + 0.3667 = 0.9414
$$

### Step 3: Calculate $$\sin(\beta - \alpha)$$

Using the difference identity:
$$
\sin(\beta - \alpha) = \sin\beta \cos\alpha - \cos\beta \sin\alpha
$$
$$
\sin(\beta - \alpha) = (0.7530)(0.487) - (0.658)(0.8734) \approx 0.3668 - 0.5747 = -0.2079
$$

### Final Answers

**(a)** $$\sin(\alpha + \beta) = 0.9414$$

**(b)** $$\sin(\beta - \alpha) = -0.2079$$
$$ \sin(\alpha + \beta) = 0.9414 $$ $$ \sin(\beta - \alpha) = -0.2079 $$

If and with both angles’ terminal rays in Quadrant-1, find the values of
(a)

(b)

Your answers should be accurate to 4 decimal places.
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If and with both angles’ terminal rays in Quadrant-1, find the values of (a) (b) Your answers should be accurate to 4 decimal places. Submit Question