Solve the equation 6 cos x^2+sin x=4 if 0 is less than or equal to x is less than or equal to pi
6 cos ^ 2 ( x ) + sin ( x ) = 4
Then :
6 * [ 1 - sin ^ 2 ( x ) ] + sin ( x )= 4
6 - 6 sin ^ 2 ( x ) + sin ( x ) = 4
- 6 sin ^ 2 ( x ) + sin ( x ) + 6 - 4 = 0
- 6 sin ^ 2 ( x ) + sin ( x ) + 2 = 0
Substitution :
sin ( x ) = u
- 6 u ^ 2 + u + 2 = 0
The exact solutions are :
u = 2 / 3
and
u = - 1 / 2
OR
sin ( x ) = 2 / 3
and
sin ( x ) = - 1 / 2
sin ^ - 1 ( 2 / 3 ) = 0.729728 radians
sin ^ - 1 ( - 1 / 2 ) = - pi / 6 radians
The period of sine function is 2 pi.
If :
0 ¡Ü x
Solution are :
x = 2 n pi + ( - pi / 6 ) =
2 n pi - pi / 6
where n is an integer
P.S.
If you don't know how to solve equation :
- 6 u ^ 2 + u + 2 = 0
Then in google type:
quadratic equation online
When you see list of results click on:
Free Online Quadratic Equation Solver:Solve by Quadratic Formula
When page be open in rectangle type:
- 6 u ^ 2 + u + 2 = 0
and click option: solve it
You will see solution step-by step
¡Ü
mean less than or equal to
To solve the equation 6cos(x^2) + sin(x) = 4, we need to find the values of x that satisfy the equation. Here is how you can approach solving this equation:
1. Combine like terms: Move the sin(x) term to the left side of the equation by subtracting sin(x) from both sides:
6cos(x^2) = 4 - sin(x)
2. Make use of the Pythagorean identity: Recall that sin^2(x) + cos^2(x) = 1. We can rewrite the equation by replacing cos(x^2) with sqrt(1 - sin^2(x)). This allows us to eliminate the cos(x^2) term:
6sqrt(1 - sin^2(x)) = 4 - sin(x)
3. Square both sides of the equation: Square both sides of the equation to eliminate the square root:
(6sqrt(1 - sin^2(x)))^2 = (4 - sin(x))^2
36(1 - sin^2(x)) = (4 - sin(x))^2
4. Simplify the equation: Distribute the squares and simplify both sides of the equation:
36 - 36sin^2(x) = 16 - 8sin(x) + sin^2(x)
5. Rearrange the equation: Move all terms to one side of the equation to form a quadratic equation:
37sin^2(x) - 8sin(x) - 20 = 0
6. Solve the quadratic equation: Factor the quadratic equation or use the quadratic formula to find the values of sin(x). Once you have the values of sin(x), you can find the corresponding values of x by taking the inverse sine (arcsin) of sin(x).
By following these steps, you can solve the given equation 6cos(x^2) + sin(x) = 4 for values of x between 0 and π.