]> Trigonometric Functions and Calculus

Trigonometric Functions and Calculus

11.1: Limits of Trigonometric Functions

While Sine and Cosine are continuous for all real numbers, the remaining functions are only continuous on their domains. Thus, if you are finding a limit of a straightforward trigonometric function for a value of x that is in the domain, then it's easy—just plug in!

When the function involves more than just a trigonometric function (ex: $y = \frac{sin x}{x}$ ), you'll probably want to investigate the graph—otherwise, it usually takes some calculus to actually find the limit!

Examples

[1.] Find $lim_{x \to \frac{π}{3}} (\sin x)$ .

Easy! Just plug in. $lim_{x \to \frac{π}{3}} (\sin x) = \sin (\frac{π}{3}) = \frac{\sqrt{3}}{2}$ .

[2.] Find $lim_{x \to 0} [x \cdot \sin (\frac{1}{x})]$ .

Well, let's look at the graph.

An Image

Yikes! Let's close in around x = 0 and see what's going on.

An Image

Looks like the limit is zero.

11.2: Derivatives of Trigonometric Functions

I'll spare you the proofs—here are the derivatives of all six trigonometric functions.

$\frac{d}{d x} [sin x] = cos x$ , $\frac{d}{d x} [cos x] = - sin x$ , $\frac{d}{d x} [tan x] = {sec}^{2} x$ , $\frac{d}{d x} [cot x] = {- csc}^{2} x$ , $\frac{d}{d x} [sec x] = (sec x) (tan x)$ , $\frac{d}{d x} [csc x] = - (csc x) (cot x)$ .

And watch out for that chain rule!

Examples

[3.] Find the derivative of y = x + sin x.

Easy! y′ = 1 + cos x.

[4.] Find the derivative of $y = \frac{cos x}{x}$ .

Quotient rule! $y^{'} = \frac{x (- sin x) - sin x}{x^{2}} = \frac{- sin x (x + 1)}{x^{2}}$ .

[5.] Find the derivative of $y = e^{tan x}$ .

$y^{'} = e^{tan x} \cdot {sec}^{2} x$ .

[6.] Find the derivative of y = ln(sec x) ( $x \in (\frac{- π}{2}, \frac{π}{2})$ ).

$y^{'} = \frac{1}{sec x} \cdot (sec x) (tan x)$ .

11.4: Inverse Trigonometric Function Derivatives

Formulas

These can be derived through implicit differentiation…but we won't go there.

$\frac{d}{d x} [arcsin f (x)] = \frac{f^{'} (x)}{\sqrt{1 - f^{2} (x)}}$

$\frac{d}{d x} [arccos f (x)] = \frac{- f^{'} (x)}{\sqrt{1 - f^{2} (x)}}$

$\frac{d}{d x} [arctan f (x)] = \frac{f^{'} (x)}{1 + f^{2} (x)}$

$\frac{d}{d x} [arccot f (x)] = \frac{- f^{'} (x)}{1 + f^{2} (x)}$

$\frac{d}{d x} [arcsec f (x)] = \frac{f^{'} (x)}{| f (x) | \sqrt{f^{2} (x) - 1}}$

$\frac{d}{d x} [arccsc f (x)] = \frac{- f^{'} (x)}{| f (x) | \sqrt{f^{2} (x) - 1}}$

Examples

[7.] Find $f^{'} (x)$ if $f (x) = arcsin (2 x)$ .

$f^{'} (x) = \frac{2}{\sqrt{1 - 4 x^{2}}}$

[8.] Find $g^{'} (x)$ if $g (x) = arctan (cos x)$ .

$g^{'} (x) = \frac{- sin x}{1 + {cos}^{2} (x)}$

Page last validated 2010-08-15