# At a Glance - Trigonometric Functions

The **trigonometric functions** are the functions used often in trigonometry:

sin(*x*), cos(*x*), tan(*x*), csc(*x*), sec(*x*), and cot(*x*).

We'll find the derivatives of some of these now, and some we'll leave until we've learned more rules.

Start with *f*(*x*) = sin(*x*). While we *could* calculate the derivative of this function using the limit definition, we'll do what we did with the logarithmic function and use pictures.

The graph of *f*(*x*) = sin(*x*) looks like this:

Wherever the function hits a maximum or minimum value, it has a horizontal tangent line:

Wherever there's a horizontal tangent line, the derivative is zero:

Estimating *f* ' (0) = 1, *f ' *(π) = -1, and *f '*(2π) = 1 (Exercise: estimate these derivatives using tables.) we find the following rough graph of the derivative:

Since we already know the answer, we'll connect the dots.:

And ... da da da dum... the derivative of *f*(*x*) = sin(*x*) is

*f ' *(*x*) = cos(*x*).

Now look at *f*(*x*) = cos(*x*):

Again, we can find all the places the derivative is zero by finding all the maxima and minima of *f*:

Estimating with tables we find *f ' *(0) = 0, *f ' *(π/2) = -1, and *f ' *(3π/2) = -1 (Exercise: estimate these derivatives using tables):

Filling in the dots, we find

The derivative of *f*(*x*) = cos(*x*) is

*f ' *(*x*) = -sin(*x*).

**Be Careful:** To remember that the derivative of sin(*x*) is +cos(*x*), and the derivative of cos(*x*) is -sin(*x*), look at the graphs. The graph of sin(*x*) starts by increasing, so its derivative must be positive at first:

The graph of cos(*x*) starts by decreasing, so its derivative must start out negative:

The other trig functions can all be written as quotients involving the sine and/or cosine functions:

To find the derivatives of these functions, first we need to know how to find the derivatives of quotients.

Deriving inverse trig functions, such as arcsin(*x*) and arccos(*x*), requires knowledge of derivatives of inverse functions in general. This uses the chain rule, which we'll discuss later.

#### Exercise 1

What is the largest the slope of *f*(*x*) = sin(*x*) ever gets? The smallest?

#### Exercise 2

What is the largest the slope of *f*(*x*) = cos(*x*) ever gets? The smallest?

#### Exercise 3

Let *f*(*x*) = -sin(*x*). Based on the discussion in this section, what is a reasonable formula for *f ' *(*x*)?

#### Exercise 4

Let *f*(*x*) = -cos(*x*). Based on the discussion in this section, what is a reasonable formula for *f ' *(*x*)?

#### Exercise 5

Find the derivative of the function. Be sure to use correct notation.

*f*(*x*) = sin(*x*)

#### Exercise 6

Find the derivative of the function. Be sure to use correct notation.

*g*(*z*) = cos(*z*)

#### Exercise 7

Find the derivative of the function. Be sure to use correct notation.

*h*(*y*) = -sin(*y*)

#### Exercise 8

Find the derivative of the function. Be sure to use correct notation.

*r*(*k*) = -cos(*k*)