Question

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

Answer 1

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

Answer 2

¡Ü

mean less than or equal to