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.