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Rewriting Trigonometric Functions

Tags
Cegep1
Mathematics
Word count
299 words
Reading time
2 minutes

cos(arcsin(x+1))

Let θ=arcsin(x+1)sinθ=x+1Let c=adjacent side of θ12=c2+(x+1)2c2=1(x+1)2c=±1(x+1)2Note that cosθ0 & cosθ=c1=cc>0cos(arcsin(x+1))=cosθ=c=1(x+1)2

sin(2arccos(ex))

Let θ=arccos(ex)cosθ=exLet c=opposite side of θ12=c2+(ex)2c2=1e2xc=±1e2xNote that sinθ0 and sinθ=c1=cc>0sin(2arccos(ex))=sin(2θ)=2sinθcosθ=2cex=21e2xex

sec(arctan(1x))

Let θ=arctan(1x)tanθ=1xThus 1= opposite side of θx=adjacent side of θLet c=hypotenuse of θPer Pythagorean Theorem, c=12+x2Since secθ>0 and secθ=cx,cx>0Note that x0 because 1x is definedThus, x>0c>0 & x<0c<0sec(arctan(1x))=secθ=cx={1+x2xx>01+x2xx<0

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