Question

8sin^3x+4cos^2x=1+6sinx

Answer 1

Решение смотрите в приложении

Answer 2

To solve the equation 8sin^3x + 4cos^2x = 1 + 6sinx, we can rewrite it in terms of sin(x) only. Here's the step-by-step solution:

1. Recall the trigonometric identity: sin^2(x) + cos^2(x) = 1. Rearranging this identity, we have cos^2(x) = 1 - sin^2(x).

2. Substitute cos^2(x) in the original equation: 8sin^3(x) + 4(1 - sin^2(x)) = 1 + 6sin(x).

3. Simplify the equation: 8sin^3(x) + 4 - 4sin^2(x) = 1 + 6sin(x).

4. Move all terms to one side of the equation: 8sin^3(x) - 4sin^2(x) - 6sin(x) + 3 = 0.

5. Factor out a sin(x): sin(x)(8sin^2(x) - 4sin(x) - 6) + 3 = 0.

6. Simplify further: 8sin^2(x) - 4sin(x) - 6 = 0.

7. Now, we can solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, we will use the quadratic formula: sin(x) = (-(-4) ± sqrt((-4)^2 - 4(8)(-6))) / (2(8)).

8. Simplify the quadratic formula: sin(x) = (4 ± sqrt(16 + 192)) / 16. sin(x) = (4 ± sqrt(208)) / 16.

9. Further simplification: sin(x) = (1 ± sqrt(13)) / 4.

Therefore, the equation 8sin^3x + 4cos^2x = 1 + 6sinx has two possible solutions for sin(x):

1. sin(x) = (1 + sqrt(13)) / 4.
2. sin(x) = (1 - sqrt(13)) / 4.

Please note that these solutions are in terms of sin(x) only. If you need to find the exact values for x, you can use the inverse sine function (sin^(-1)) to find the angles corresponding to these sin(x) values.