Final answer:
To find sinx + cosx, we use trigonometric identities to rewrite the equation sin(x) × cos(x) = 3/8. By substituting known identities, we find that sin(x) + cos(x) = (3√(2) + √(14)) / 8.
Step-by-step explanation:
Double-angle formula for sine: We can use the double-angle formula for sine, which states: sin(2x) = 2 × sin(x) × cos(x).
Rearrange for sin(x) × cos(x): Rearranging the formula for sin(x) × cos(x), we get: sin(x) × os(x) = sin(2x) / 2.
Substitute the given product: Substituting the given product of sin(x) × cos(x) = 3/8, we get: sin(2x) = 2 × (3/8) = 3/4.
Find sin(x) and cos(x) using the Pythagorean identity: Now, we can use the Pythagorean identity for sine and cosine, which states: sin²(x) + cos²(x) = 1.
Rearrange for cos²(x): Rearranging the equation for cos²(x), we get: cos²(x) = 1 - sin²(x).
Substitute the double-angle formula for sin²(x): Since we know sin(2x) = 3/4, we can find sin²(x) using the half-angle formula: sin²(x) = (1 - cos(2x)) / 2. Substituting for sin(2x), we get: sin²(x) = (1 - 3/4) / 2 = 1/8.
Find cos²(x): Substituting the calculated sin²(x) back into the equation for cos^2(x), we get: cos²(x) = 1 - 1/8 = 7/8.
Take the square root to find cos(x): Since cos(x) can be positive or negative, we need to take the square root of both sides to find two possible values: cos(x) = ±√(7/8) = ±√(7) / 2√(2).
Find sin(x) + cos(x) with the same sign: Since sin(x) × cos(x) is positive, both sin(x) and cos(x) must have the same sign. Therefore, we choose the positive value for cos(x): cos(x) = √(7) / 2√(2).
Substitute back into the double-angle formula: Using the positive value of cos(x), we can find sin(x) from the double-angle formula: sin(x) = 3 / (4 × √(2)) = 3√(2) / 8.
Finally, add sin(x) and cos(x): sin(x) + cos(x) = 3√(2) / 8 + √(7) / 2√(2) = (3√(2) + √(14)) / 8.
Therefore, sin(x) + cos(x) = (3√(2) + √(14)) / 8.