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

Step-by-step explanation:

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

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

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

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

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

$\sin {}^{2} (x) = \frac{1}{2}$

$\sin(x) = \frac{1}{ \sqrt{2} }$

Take the arc Sine of the function

$arcsin( \frac{1}{ \sqrt{2} } ) = \frac{\pi}{4}$

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

$\sin( \frac{\pi}{4} ) = \frac{1}{ \sqrt{2} }$

so using the identity

$\sin( \frac{\pi}{4} + \pi ) = \sin( \frac{5\pi}{4} )$

So we have two answers.

$( \frac{\pi}{4} )$

$( \frac{5\pi}{4} )$