Question

\[\frac{ \sin ^{2}x-\sin x-2 }{ \sin x-2 }? \] what is equivalent

Answer 1

The numerator is a quadratic equation in \(\sin x\).
Try factoring the numerator.

Answer 2

If factoring the numerator directly looks confusing, then use a substitution.
Let \(u = \sin x\)
Then the numerator is \(u^2 - u - 2\)
Can you factor that quadratic equation in u?

Answer 3

\(\bf \cfrac{ \sin ^{2}(x)-\sin (x)-2 }{ \sin (x)-2 }\implies \cfrac{ [{\color{brown}{ sin(x)}}]^2-{\color{brown}{ sin (x)}}-2 }{ {\color{brown}{ sin (x)}}-2 }\implies
\cfrac{ [{\color{brown}{ u}}]^2-{\color{brown}{ u}}-2 }{ {\color{brown}{ u}}-2 }\)

notice what mathstudent55 suggested

Answer 4

sin^2x?

Answer 5

sin x^2 - 1?

Answer 6

No.
Can you factor \(u^2 - u - 2 \) ?

Answer 7

no?

Answer 8

If you are asking if that is a final answer, the answer is no.

Answer 9

ok so what is the answer?

Answer 10

I can give you the answer, but that only shows that I know ho to solve it, not that you learned anything.
I am trying to help you by breaking down this problem into simple steps, but you haven't answered any of the questions I asked.

Answer 11

Factor: \(u^2 - u - 2\)

Answer 12

(u-2)(u+1)

Answer 13

now what?

Answer 14

Great
Now since we substituted u for sin x, let's substitute sin x back in for u.

Answer 15

sin x -2 and sin x +1

Answer 16

This is everything we have done so far:

\(\dfrac{\sin^2 x - \sin x - 2}{\sin x - 2} \)

Let \(u = \sin x\)

\(=\dfrac{u^2 - u - 2}{\sin x - 2} \)

\(= \dfrac{(u - 2)(u + 1)}{\sin x - 2} \)

Now we substitute \(\sin x\) back in for u:

\(=\dfrac{(\sin x - 2)(\sin x + 1)}{\sin x - 2} \)

Answer 17

Now what cancels out?

Answer 18

sin x - 2

Answer 19

yeap
thus \(\bf \cfrac{ \sin ^{2}(x)-\sin (x)-2 }{ \sin (x)-2 }\implies \cfrac{ [{\color{brown}{ sin(x)}}]^2-{\color{brown}{ sin (x)}}-2 }{ {\color{brown}{ sin (x)}}-2 }\implies
\cfrac{ [{\color{brown}{ u}}]^2-{\color{brown}{ u}}-2 }{ {\color{brown}{ u}}-2 }
\\ \quad \\
\cfrac{\cancel{ ({\color{brown}{ u}}-2)}({\color{brown}{ u}}+1)}{\cancel{ {\color{brown}{ u}}-2}}\implies \cfrac{{\color{brown}{ sin(x)}}+1}{1}\implies sin(x)+1\)

Answer 20

so sin x + 1 is the answer

Answer 21

Correct

Answer 22

\(=\dfrac{(\cancel{\sin x - 2})(\sin x + 1)}{\cancel{\sin x - 2}~~1} \)

\(=\sin x + 1\)