To solve the equation 3sinxtanx = 8 for x in the interval 0 <= x <= 360, follow these steps:
1. Divide both sides of the equation by 3 to isolate sinx * tanx.
sinx * tanx = 8/3
2. Rewrite sinx * tanx as sinx * (sinx / cosx) using the identity tanx = sinx / cosx.
sinx * (sinx / cosx) = 8/3
3. Simplify the equation further by multiplying sinx:
(sin^2x) / cosx = 8/3
4. Multiply both sides of the equation by cosx to eliminate the denominator:
(sin^2x) = (8/3) * cosx
5. Rewrite sin^2x as (1 - cos^2x) using the Pythagorean identity sin^2x + cos^2x = 1:
(1 - cos^2x) = (8/3) * cosx
6. Distribute (8/3) to both terms on the right side:
1 - cos^2x = (8/3) * cosx
7. Move all terms to one side of the equation:
1 - (8/3) * cosx - cos^2x = 0
8. Rearrange the quadratic equation in standard form:
cos^2x + (8/3) * cosx - 1 = 0
9. Factor the quadratic equation:
(3cosx - 1)(cosx + 3) = 0
10. Set each factor equal to zero and solve for cosx:
3cosx - 1 = 0 or cosx + 3 = 0
11. Solve the first equation, 3cosx - 1 = 0, for cosx:
3cosx = 1
cosx = 1/3
12. Solve the second equation, cosx + 3 = 0, for cosx:
cosx = -3
13. Since cosx cannot be -3 in the interval 0 <= x <= 360, we discard that solution.
14. To find the values of x, take the inverse cosine (arccos) of cosx:
x = arccos(1/3) = 70.5º
15. Since cosine is positive in both the first and fourth quadrant, the second solution is:
x = 360º - 70.5º = 289.5º
The two solutions for x in the interval 0 <= x <= 360 are 70.5º and 289.5º.