## How to Compute the Derivative of a Trigonometric Function

Trigonometric functions, also called circular functions, are functions of angles. You’re probably already familiar with the six trigonometric functions: sin x, cos x, tan x, sec x, csc x and cot x. There’s a table that can help you figure out derivatives for the six trig functions:
d/dx sin x = cos x
d/dx csc x = -csc x cot x
d/dx cos x = – sin x
d/dx sec x = sec x tan x
d/dx tan x = sec2 x
d/dx cot x = – csc2 x
However, you might be asked (especially in beginning calculus) to find the derivative of a trig function using the definition of a derivative instead of a table. When you use the definition of a derivative, you’re actually working on a proof. In other words, if you want to prove that one function is a derivative of another, you’ll nearly always start with the definition of a derivative and end with the derivative of the trigonometric function.

Green:Cosine, Blue:Sine, Red:Tangent, Yellow:Cosecant, Magenta:Secant, Cyan:Cotan

Sample Problem: Find the derivative of a trigonometric function (sin x) using the definition of a derivative (in other words, prove that d/dx sin x = cos x:

The definition of a derivative

Step 1: Insert the function sin x into the definition of a derivative:

Step 2: Use the trigonometric identity sin(a+b)=sin a * cos B + cos a * sin B to rewrite the definition from Step 1:

Step 3: Use algebra to rewrite the formula in Step 2:

= – sin x* (0) + cos x * (1) = cos x

That’s it!

Tip: You can use the exact same technique to work out a proof for any trigonometric function. Start with the definition of a derivative and identify the trig functions that fit the bill.