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## Homework Statement

Differentiate sin(ax), cos(ax) and tan(ax) from first principles.

## Homework Equations

## The Attempt at a Solution

I have used first principles to differentiate the three expressions and have been successful until I encountered limits of some expressions in the process.

I need to find the limit as Δx tends to 0 of the following expressions.

1. sin(aΔx)/Δx

2. [cos(Δx) - 1]/Δx

3. tan(Δx)/Δx

I know some spooky proofs which use the fact that for small Δx, sin(aΔx) ≈ Δx, cos(aΔx) ≈ 1- (aΔx)

^{2}/2 and tan(aΔx) ≈ Δx. They do give the right answers and I have been told these methods would give me full marks in the exam (me being a physics student and all that crap!), but I would appreciate it if you give a full rigorous proof of the three limits. (Armed with those, the original problem is just a piece of cake.)