Tan(theta)-sin(theta)/sin(cube)(theta) When limit theta tends to zero.
asked Oct 8, 2016 in Calculus Answers by anonymous

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1 Answer

Writing t for theta, tan(t)=sin(t)sec(t),

so if t≠0 we can write: (tan(t)-sin(t))/(sin(t))^3=(sec(t)-1)/(sin(t))^2.

The expansion of cos(t)=1-t^2/2 when t is small, and sin(t)=t.

Also, sec(t)=1/cos(t)=(1-t^2/2)^-1=1+t^2/2 when t is small. Therefore sec(t)-1=t^2/2 And (sin(t))^2=t^2.

So the quotient approaches t^2/2t^2=1/2 as t approaches zero.

The limit is therefore 1/2.

answered Oct 8, 2016 by Rod Top Rated User (429,320 points)

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