Question

Would someone be willing to explain a pre-calc problem involving solving trigonometric equations?

Find all the solutions to the equation.
7 (sin^2)x - 14 sin x + 2 = -5

Answer 1

this equation combines trig and algebra.

for the moment , let y = sin(x) . This makes it easier to see:
\[ 7 \sin^2x - 14 \sin x + 2 = -5 \\
7 y^2 -14y +2 = -5\]

First step: solve for y (you get two solutions). Do you know how ?

Answer 2

Is it:
7y^2-14y+2=-5
7y^2-14y=-7
7(y^2-2y)=-7
7(y-1)(y-1)
Solutions: y=1 and y=1?

Answer 3

you have the correct factors, but your method is dubious.
The standard way is to put it in standard form, like this
7y^2-14y=-7
7 y^2 -14y +7 = 0 (add 7 to both sides
factor out 7:
7(y^2-2y+1) = 0
we can factor y^2-2y+1 into (y-1)(y-1)
so
7 (y-1)(y-1) = 0
and that means y=1

Answer 4

now remember y is short for sin x
so
y=1 means
sin x = 1

now list all x's where sin x = 1

Answer 5

y=1 at 90 degrees (pi/2). Is that what you mean by listing the x's?

Answer 6

@phi ?

Answer 7

yes, but if you want all the answers, remember the sine goes on forever. so every 2pi (or 360 degrees) later, it reaches 1 again.
so the complete answer is
\[ x= \frac{\pi}{2} + n2\pi \text{ for all integer n} \]