When you simplify:
sin(pi+x)cos((pi/2)-x)-cos(pi+x)sin(x+(3pi/2))
What's the answer cause I got -2sinxcosx, which I don't know if can simplify to -2sinx
But the answer key says -1 so. I'm not sure how to get that
You should know these identities:
sin(Ï€+x) = -sinx
e.g. sin 210° = sin(180°+30°) = -sin30°
cos(Ï€/2) - x) = sinx
e.g. cos(60) = cos(90-30) = sin30
cos(Ï€+x) = -cosx
e.g. cos 210° = -cos30°
sin(3Ï€/2 + x) = -cosx
e.g. sin 300° = sin(270+30) = -cos30
so ....
sin(pi+x)cos((pi/2)-x)-cos(pi+x)sin(x+(3pi/2))
= (-sinx)(sinx) - (-cosx)(-cosx)
= - sin^2 x - cos^2 x
= -1(sin^2 x + cos^2 x)
= -1(1)
= -1
To simplify the given expression, we need to use trigonometric identities and simplify each term separately:
First, let's expand the expression:
sin(pi + x)cos((pi/2) - x) - cos(pi + x)sin(x + (3pi/2))
Using the sum identities for sine and cosine, we can rewrite the expression as:
(sin(pi)cos(x) + cos(pi)sin(x))(cos(pi/2)cos(x) + sin(pi/2)sin(x))
- (cos(pi)cos(x) - sin(pi)sin(x))(sin(x)cos(pi/2) + cos(pi/2)sin(x + (3pi/2)))
Now, let's simplify each term:
sin(pi) = 0 and cos(pi) = -1, so the first term becomes:
0cos(x) + (-1)sin(x) = -sin(x)
cos(pi/2) = 0 and sin(pi/2) = 1, so the second term becomes:
0cos(x) + 1sin(x) = sin(x)
cos(pi) = -1 and sin(pi) = 0, so the third term becomes:
(-1)cos(x) - 0sin(x) = -cos(x)
sin(pi/2) = 1 and cos(pi/2) = 0, so the fourth term becomes:
1cos(x) + 0sin(x + (3pi/2)) = cos(x)
Substituting these simplified terms back into the expression:
-sin(x) * sin(x) - (-cos(x) * cos(x))
Using the identity sin^2(x) + cos^2(x) = 1, we can simplify further:
-(sin^2(x) + cos^2(x))
Since sin^2(x) + cos^2(x) is equal to 1, the final answer is:
-1
Therefore, the correct answer is indeed -1, and not -2sin(x) or -2sin(x)cos(x).