solve 12cos^2X=8+13sinX between the window of -360 ¡Ü x ¡Ü 360

change cos^2 to 1-sin^2, then arrange the terms as a quadratic, and factor.

so would it be (3sinx-1)(4sinx+4)=0?

then solve for x...

after i solve for x then that is my answer? so x=1/3,-1

To solve the equation 12cos^2X = 8 + 13sinX within the given window -360 ≤ x ≤ 360, we can follow these steps:

Step 1: Rewrite the equation in terms of a single trigonometric function.
Using the identity cos^2X = 1 - sin^2X, we can rewrite the equation as follows:
12(1 - sin^2X) = 8 + 13sinX

Step 2: Simplify the equation.
Distribute 12 to both terms inside the parentheses:
12 - 12sin^2X = 8 + 13sinX

Rearrange the equation to match the quadratic form:
12sin^2X + 13sinX - 4 = 0

Step 3: Solve the quadratic equation.
Let's solve the quadratic equation using factoring, completing the square, or the quadratic formula. However, this equation does not have simple factorization or perfect square trinomial form. So, let's use the quadratic formula:
sinX = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 12, b = 13, and c = -4.

sinX = (-13 ± √(13^2 - 4(12)(-4))) / 2(12)
sinX = (-13 ± √(169 + 192)) / 24
sinX = (-13 ± √361) / 24

Step 4: Find the values of sinX.
Now, let's find the possible values of sinX by evaluating the ±√361.

sinX = (-13 + √361) / 24 or sinX = (-13 - √361) / 24

sinX = (-13 + 19) / 24 or sinX = (-13 - 19) / 24
sinX = 6/24 or sinX = -32/24
sinX = 1/4 or sinX = -4/3

Step 5: Find the corresponding values of X.
To find the corresponding values of X, we can use the inverse sine function (sin^-1) and evaluate sin^-1(1/4) and sin^-1(-4/3).

X = sin^-1(1/4) or X = sin^-1(-4/3)

Using a calculator or trigonometric table, you can approximate that:
X ≈ 14.48°, 75.52°, 165.52°, 195.52°, 284.48°, 345.52°

These are the approximate values of X within the given window -360 ≤ x ≤ 360 that satisfy the equation 12cos^2X = 8 + 13sinX.