Simplify to a single term
Sin(540-x).sin(-x) -cos(180-x).sin(90+x)
To simplify the expression Sin(540-x).sin(-x) -cos(180-x).sin(90+x), we can use trigonometric identities to rewrite the terms:
1. Recall that sin(-x) = -sin(x) and cos(180-x) = -cos(x). Applying these identities:
Sin(540-x).sin(-x) -cos(180-x).sin(90+x) becomes:
= Sin(540-x)(-sin(x)) - (-cos(x)).sin(90+x)
2. Next, let's simplify the sin(90+x) term. Recall that sin(A+B) = sin(A)cos(B) + cos(A)sin(B). Applying this:
sin(90+x) = sin(90)cos(x) + cos(90)sin(x)
= cos(x)
Substituting this back into the expression:
= Sin(540-x)(-sin(x)) - (-cos(x)).cos(x)
= -Sin(540-x)sin(x) + cos^2(x)
So, the simplified expression is -Sin(540-x)sin(x) + cos^2(x).
To simplify the expression, let's break it down step by step:
1. Recall the trigonometric identity:
sin(-x) = -sin(x)
2. Apply the above identity to the expression:
Sin(540-x).sin(-x) -cos(180-x).sin(90+x)
= Sin(540-x).(-sin(x)) -cos(180-x).sin(90+x)
3. Use the angle addition/subtraction identity for sin (sin(a + b) = sin(a)cos(b) + cos(a)sin(b)) to expand the expression further:
= Sin(540).cos(x) - Sin(x).sin(x) - cos(180).sin(90).cos(x) - cos(180).cos(90).sin(x)
4. Simplify the trigonometric functions:
Since Sin(540) = 0, Sin(540).cos(x) = 0. Similarly, sin(90) = 1 and cos(90) = 0. Therefore, the expression becomes:
= 0 - Sin(x).sin(x) - 0 - 0
= -Sin^2(x)
So, the simplified expression is -Sin^2(x).
Play with the identities
sin(540 - x) = sin(180 - x) = sinx , (540° is one rotation + 180°)
sin(-x) = sinx
cos(180-x) = -cosx
sin(90+x) = sin90cosx + cos90sinx = cosx + 0 = cosx
Sin(540-x).sin(-x) -cos(180-x).sin(90+x)
= (sinx)(sinx) - (-cosx)(cosx)
= sin^2 x + cos^2 x
= 1