# trigonometry

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if cosθ=2/3 and tanθ<0 find the exact value

sin(θ-5π/3)

sin2θ

cos2θ

cos(θ/2)

• trigonometry -

Solution:Step1First replace x with (x-Î”x) in the given funicton f(x) to obtainf(x+Î”x) = 2(x+Î”x)+1 = 2x + 2Î”x + 1 Step2Rewrite f(x-Î”x) as (2x + 2Î”x + 1) and f(x) as (2x + 1) into the definition of the derivative limit to obtain: lim [(2x + 2Î”x + 1) - (2x+1)] / Î”xÎ”x-> 0 Step 3Combine like terms on the numerator and the reduced expression becomes lim [2Î”x] / Î”xÎ”x-> 0 Step 4Cancel out Î”x from both numerator and denominator and finally you get the derivative Derivative of f (x) = 2x+1 = 2 -Note there was no need to substitute Î”x with 0 since it was canceled out.

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