SUNY Geneseo Department of Mathematics

Derivatives of Trigonometric Functions

Thursday, September 26

Math 221 06
Fall 2019
Prof. Doug Baldwin

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Previous Lecture

Misc

SI session this afternoon 3:00 - 4:30, Fraser 104.

Questions?

Derivative of Sine (and Cosine)

Section 3.5

Key idea: proof via the limit definition of derivative, with some apt uses of the squeeze theorem to find certain crucial limits.

Use these ideas to show that the derivative of cos x is -sin x.

Start with the limit definition applied to cosine:

Derivative of cosine is limit as h goes to 0 of cosine x plus h minus cosine x all over h

Then use a trigonometric identity to expand cos(x+h):

Cosine x plus h is cosine x times cosine h plus sine x times sine h

Then rearrange to separate terms involving x and terms involving h (rationale: any terms that involve x are just constants as far as limits as h approaches 0 are concerned, and so will be easy to evaluate; remaining terms involving h will hopefully be evaluable via limit laws or similar).

Limit becomes limit of product of cosines minus limit of product of sines

Finally, the limits with h in them can be evaluated via the squeeze theorem. One goes to 0 and the other to 1, leaving a simple expression for the derivative.

Limit of cosine h minus 1 all over h is 0, limit of sin h over h is 1

Derivatives of Other Trig Functions

Show that the derivative of sec x is sec x tan x.

Key idea: proof via the quotient rule, which is the crux of all derivations for trig functions besides sine and cosine.

secx = 1 / cosx

Apply quotient rule to 1 over cosine x to get sine x over cosine x times 1 over cosine x

Next

Key trigonometric antiderivatives, maybe examples of applying trigonometric differentiation rules.

Then the chain rule.

Read section 3.6 of the textbook.

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