Question

3sin x cos x = 1

Answer 1

3sin x cos x = 1

3/2 sin 2x=1

sin 2x=2/3

2x=(-1)^k * arcisn (2/3)+pi*k, k є Z

x=1/2*(-1)^k * arcsшn (2/3) +pi/2*k, k є Z

Answer 2

To solve the equation 3sin(x)cos(x) = 1, we can use trigonometric identities to simplify it and find the solutions. Let's go through the steps:

1. Start with the equation: 3sin(x)cos(x) = 1.

2. Apply the double angle formula for sine: sin(2x) = 2sin(x)cos(x).

3. Rewrite the equation using the double angle formula: 2sin(x)cos(x) = 1.

4. Divide both sides of the equation by 2: sin(x)cos(x) = 1/2.

5. Apply the double angle formula for cosine: cos(2x) = cos^2(x) - sin^2(x).

6. Substitute the identities: cos(2x) = cos^2(x) - (1 - cos^2(x)).

7. Simplify the equation: cos(2x) = 2cos^2(x) - 1.

8. Rewrite the equation: 2cos^2(x) - cos(2x) - 1 = 0.

Now, we have transformed the equation into a quadratic form. Let's solve it:

9. Set the equation equal to zero: 2cos^2(x) - cos(2x) - 1 = 0.

10. Let's introduce a substitution to simplify the equation. Let u = cos(x). Now, we have:

2u^2 - cos(2x) - 1 = 0.

11. Use the double angle formula for cosine: cos(2x) = 2cos^2(x) - 1.

Substituting this into the equation, we get:

2u^2 - (2u^2 - 1) - 1 = 0.

12. Simplify the equation: 2u^2 - 2u^2 + 1 - 1 - 1 = 0.

This simplifies to: 0 = 0.

The equation simplifies to 0 = 0, which is true for all values of u (or cos(x)). This means that the original equation 3sin(x)cos(x) = 1 is satisfied by all possible values of x.

In conclusion, the solution to the equation 3sin(x)cos(x) = 1 is x can be any real number.