Question

√2 sin x-√2 cos x=√3

Answer 1

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Answer 2

To solve the equation √2 sin(x) - √2 cos(x) = √3, you can follow these steps:

1. Start by factoring out √2 from both terms on the left side of the equation: √2 (sin(x) - cos(x)) = √3

2. Divide both sides of the equation by √2 to isolate the expression inside the parentheses: sin(x) - cos(x) = √3 / √2

3. Simplify the right side by rationalizing the denominator (since √2 = √(2/2) = 1/√2): sin(x) - cos(x) = (√3 / √2) * (√2 / √2) sin(x) - cos(x) = (√6 / 2)

4. Now, you have the equation: sin(x) - cos(x) = √6 / 2

5. To solve for x, you can use trigonometric identities or techniques. One common approach is to rewrite sin(x) and cos(x) in terms of sine and cosine of a common angle using the angle sum identities: sin(x) = sin(45°)sin(x) + cos(45°)cos(x) cos(x) = cos(45°)sin(x) - sin(45°)cos(x)

6. Substitute these expressions into the equation: [sin(45°)sin(x) + cos(45°)cos(x)] - [cos(45°)sin(x) - sin(45°)cos(x)] = √6 / 2

7. Simplify the equation by canceling out terms: (sin(45°)sin(x) + cos(45°)cos(x)) - (cos(45°)sin(x) - sin(45°)cos(x)) = √6 / 2

   [sin(45°)sin(x) + cos(45°)cos(x) - cos(45°)sin(x) + sin(45°)cos(x)] = √6 / 2

8. Combine like terms: [sin(45°)sin(x) - cos(45°)sin(x)] + [cos(45°)cos(x) + sin(45°)cos(x)] = √6 / 2

9. Factor out sin(x) and cos(x) on both sides: sin(x)[sin(45°) - cos(45°)] + cos(x)[cos(45°) + sin(45°)] = √6 / 2

10. Evaluate the trigonometric values for sin(45°) and cos(45°): sin(45°) = cos(45°) = 1 / √2

11. Substitute these values into the equation: sin(x)[1/√2 - 1/√2] + cos(x)[1/√2 + 1/√2] = √6 / 2

12. Simplify further: sin(x)[0] + cos(x)[√2/√2] = √6 / 2 cos(x) = √6 / 2

13. Now, solve for x by taking the arccosine of both sides: x = arccos(√6 / 2)

14. You can use a calculator to find the numerical value of x: x ≈ 30° or x ≈ 330° (since the cosine function has a period of 360°)

So, the solutions for x are approximately x ≈ 30° and x ≈ 330°.