solve for x
6tanx - 4sinx = 1
i tried it but got a quartic formula :S
16sin^4x + 8sin^3x + 37sin^2x - 36 = 0
This is a "nasty" problem and it does not simplify easily.
I don't know how you got your quartic equation.
I rewrote it as 6tanx = 1 + 4sinx
then graphed f(x) = 6tanx and
g(x) = 1+ 4sinx
There are two intersection points, one in quadrant I and another in quadrant III.
After a few iterations, I got x= 23.27º to come pretty close.
This is way beyond any highschool level I know, where did you get that question ?
I got the quartic by moving the sin to the right, squaring both sides, then multiplying everything by sin^2 and then changing the cos to sin.
My friend gave me this question, and he got it from his virtual school for gr 12 advanced functions lol.
To solve the equation 6tanx - 4sinx = 1, we can simplify it by using the trigonometric identity tanx = sinx/cosx.
Let's start by substituting tanx = sinx/cosx into the equation:
6(sinx/cosx) - 4sinx = 1
Next, multiply both sides of the equation by cosx to eliminate the denominator:
6sinx - 4sinx * cosx = cosx
Now, we want to somehow get all the trigonometric terms on one side of the equation and set the equation equal to zero. To do this, let's move all the terms to the left side:
6sinx - 4sinx * cosx - cosx = 0
Combine like terms:
(6sinx - cosx) - 4sinx * cosx = 0
Now, the equation has taken a form where we have a product of two terms (4sinx * cosx) and another term (6sinx - cosx). We can factor out sinx from the first two terms:
sinx(6 - 4cosx) - cosx = 0
Now, we can factor out cosx from the second two terms:
sinx(6 - 4cosx) - cosx(1) = 0
Taking out the common factors:
sinx(6 - 4cosx) - cosx = 0
Now, set each factor equal to zero and solve for x:
sinx = 0 or 6 - 4cosx = 0
For sinx = 0, we know that sinx = 0 when x = 0π, 1π, 2π, etc.
For 6 - 4cosx = 0, solve for cosx:
6 - 4cosx = 0
4cosx = 6
cosx = 6/4
cosx = 3/2
However, the value of cosx cannot exceed 1 or be less than -1. Therefore, the equation 6 - 4cosx = 0 has no real solutions.
Therefore, the solutions for x are x = 0π, 1π, 2π, etc.