How should i go about solving the equation: 7sin(x)+5=2cos^2(x)

Question

jim_thompson5910 · Answer

First start with the pythagorean identity and isolate cos^2

$$\Large \sin^2(\theta)+\cos^2(\theta) = 1$$

$$\Large \cos^2(\theta) = 1-\sin^2(\theta)$$

anonymous · Answer

then distribute the 2?

jim_thompson5910 · Answer

Yes, to get 

$$\Large 7\sin(x) + 5 = 2-2\sin^2(x)$$

jim_thompson5910 · Answer

Now let $\Large z = \sin(x)$

Squaring both sides gives you $\Large z^2 = \sin^2(x)$

jim_thompson5910 · Answer

Because of those two assignments, we can replace all copies of sin(x) with z to go from

$$\Large 7\sin(x) + 5 = 2-2\sin^2(x)$$

to

$$\Large 7z + 5 = 2-2z^2$$

anonymous · Answer

then solve normally and throw the trig back in?

jim_thompson5910 · Answer

Exactly. You solve either by factoring, completing the square, or the quadratic formula

Once you have your solutions in terms of z, you plug z = sin(x) back in and solve for x (in each equation)

anonymous · Answer

thank you so much

jim_thompson5910 · Answer

you're welcome