sin(pi+x) cos(pi/2-x) - cos(pi+x) sin(x+3pi/2)
draw it
sin (pi+x) = -sin x
cos (pi/2 - x) = sin x
cos(pi+x) = -cos x
sin(1.5pi + x ) = -cos x
so
-sin x* sin x - (-cos x)* (- cos x)
= - (sin^2 x + cos^2 x)
= -1
Use trigonometric identities:
sin ( π + x ) = - sin x
cos ( π / 2 - x ) = sin x
cos ( π + x ) = - cos x
sin ( x + 3 π / 2 ) = - cos x
So
sin ( π + x ) ∙ cos ( π / 2 - x ) - cos ( π + x ) ∙ sin ( x + 3 π / 2 ) =
- sin x ∙ sin x - ( - cos x ) ∙ ( - cos x ) =
- sin² x - cos² x = - ( sin² x + cos² x ) = - 1
To evaluate the expression sin(pi+x) cos(pi/2-x) - cos(pi+x) sin(x+3pi/2), we can simplify each term separately and then perform the appropriate substitutions.
1. Simplifying sin(pi+x):
Using the angle sum identity, sin(a+b) = sin(a)cos(b) + cos(a)sin(b), we have:
sin(pi+x) = sin(pi)cos(x) + cos(pi)sin(x) = 0*cos(x) + (-1)sin(x) = -sin(x).
2. Simplifying cos(pi/2-x):
Using the angle subtraction identity, cos(a-b) = cos(a)cos(b) + sin(a)sin(b), we have:
cos(pi/2-x) = cos(pi/2)cos(x) + sin(pi/2)sin(x) = 0*cos(x) + 1*sin(x) = sin(x).
3. Simplifying cos(pi+x):
Using the angle sum identity, cos(a+b) = cos(a)cos(b) - sin(a)sin(b), we have:
cos(pi+x) = cos(pi)cos(x) - sin(pi)sin(x) = (-1)cos(x) - 0*sin(x) = -cos(x).
4. Simplifying sin(x+3pi/2):
Using the angle sum identity, sin(a+b) = sin(a)cos(b) + cos(a)sin(b), we have:
sin(x+3pi/2) = sin(x)cos(3pi/2) + cos(x)sin(3pi/2) = sin(x)*0 + cos(x)(-1) = -cos(x).
Now, let's substitute these simplified expressions back into the original equation:
-sin(x) * sin(x) - (-cos(x)) * (-cos(x)) = -sin^2(x) - cos^2(x).
Using the Pythagorean identity, sin^2(x) + cos^2(x) = 1, we can rewrite the equation as:
-1.
Therefore, the value of sin(pi+x) cos(pi/2-x) - cos(pi+x) sin(x+3pi/2) is -1.