Superbowl Express
How do I express this trig expression as an alg expression?
sin(arctan2x-arccosx)
I've been stuck on this for a while, I looked at the example problems in my book but I can't figure this out. My mom said if I don't finish my HW I can't go to the superbowl party up the block 
plz help me!! Free 10 points ;D
sin(arctan2x-arccosx)
Let y = arctan 2x; tan y = 2x;
Since Tan y = 2x
This implies a right triangle with angle y, has an opposite side of length 2x and adjacent side of length 1, and a hypothenuse of sqrt(4x^2 + 1).
Sin y = 2x/sqrt(4x^2 + 1)
Cos y = 1/sqrt(4x^2 + 1)
Let z = arccos x; cos z = x
For angle z, the adjacent side = x, hypothenuse = 1; opposite side = sqrt(1 - x^2)
Sin z = sqrt(1 - x^2)
Tan z = sqrt(1 - x^2)/x
sin(arctan2x-arccosx) = sin(y - z)
Using trig identity (sum of 2 angles)
Sin (y - z) = sin y cos z - cos y sin z
= (2x/sqrt(4x^2 + 1))(x) - (1/sqrt(4x^2 + 1))(sqrt(1 - x^2))
= 2x^2/sqrt(4x^2 + 1) - (sqrt(1 - x^2))/(sqrt(4x^2 + 1))
= (2x^2 - sqrt(1 - x^2))/(sqrt(4x^2 + 1)
<< that's your algebraic expression !! - it is kindda long... >>
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