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# Trigonometry Derivatives

While you may know how to take the derivative of a polynomial, what happens when you need to take the derivative of a trig function? What IS the derivative of a sine?

Luckily, the derivatives of trig functions are simple -- they're other trig functions! For example, the derivative of sine is just cosine:

The rest of the trig functions are also straightforward once you learn them, but they aren't QUITE as easy as the first two.

**Derivatives of Trigonometry Functions**

sin'(x) = cos(x)

cos'(x) = -sin(x)

tan'(x) = sec2(x)

sec'(x) = sec(x)tan(x)

cot'(x) = -csc2(x)

csc'(x) = -csc(x)cot(x)

### How did we calculate these functions?

The characteristic trigonometric identity to recall in calculus is this:

It says that the **derivative of sine is cosine**, and the **derivative of cosine is negative sine**.

From these we may derive the rest of the derivatives, via the Quotient and Product rules. See if you can follow along as we derive them!

### Derivative of Secant

Remember that the secant is the inverse of cosine -- it's 1/cos(x). Rewrite it as such, and find the derivative:

All we did was think of 1/cos(x) as cos(x) to the -1 power. Finally we simplified a bit by converting sin/cos to tan and the remaining 1/cos term to sec.

### Derivative of Cosecant

Just like secant, except with sine instead of cosine. That's because cosecant is 1/sin!

### Derivative of Tangent

For tangent, let's rewrite tangent as sin*sec. Remember that tangent is sin/cos, which is the same as sin*sec. It's often easier to differentiate a product, so we'll use that version and substitute the derivative of sec from above.

### Derivative of Cotangent

### To sum up:

So, that's how we mathematically derive the derivative of each trig function. Let's step back though a visually demonstrate the relationship between sine and cosine, and the neat fact that they are each other's derivatives (plus a negative sign when appropriate). Take a look at this graphic for an illustration of what this means. At the first point (around x=2*pi), the cosine isn't changing. You can see that the sine is 0, and since negative sine is the rate of change of cosine, cosine would be changing at a rate of -0.

At the second point I've illustrated (x=3*pi), you can see that the sine is decreasing rapidly. This makes sense because the cosine is negative. Since cosine is the rate of change of sine, a negative cosine means the sine is decreasing.