Question

Solve for x the equation 6cos^2 x+4sin^2 x=5.Forex 2 question 6 cos square x + 4 sin square x is equal to 5 ​

Answer 1

Explanation:

`6 \cos {}^(2) (x) + 4 \sin {}^(2) (x) = 5`

`6(1 - sin {}^(2) x) + 4 \sin {}^(2) (x) = 5`

`6 - 6 \sin {}^(2) (x) + 4 \sin {}^(2) (x) = 5`

`6 - 2 \sin { }^(2) (x) = 5`

`- 2 \sin {}^(2) (x) = - 1`

`\sin {}^(2) (x) = (1)/(2)`

`\sin(x) = (1)/( √(2) )`

Take the arc Sine of the function

`arcsin( (1)/( √(2) ) ) = (\pi)/(4)`

Finally, Sine is a periodic function so it has two answers within the interval (0, 2 pi)

Use the identity

`\sin(x) = \sin(x + \pi)`

We know that

`\sin( (\pi)/(4) ) = (1)/( √(2) )`

so using the identity

`\sin( (\pi)/(4) + \pi ) = \sin( (5\pi)/(4) )`

So we have two answers.

`( (\pi)/(4) )`

`( (5\pi)/(4) )`