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

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)** \(\sin(\beta - \alpha) = -0.2079\)