# Homework Help: Calc midterm tomorrow

1. Oct 24, 2005

### Physics is Phun

Ok, i need an easy way to remember all the trig derivatives. that's reiprical trig, and all the inverses of them aswell.
Is there some easy way where i only have to remember a few of them, and i can just figure out the rest from that?

thanks

2. Oct 24, 2005

### amcavoy

You don't need to memorize them. If you know the derivatives of sine and cosine, then you can get everything else from the product or quotient rules. For the inverses, either use the inverse rule for derivatives or do something like this:

$$\frac{d}{d\theta}\tan\left(\arctan{\theta}\right)=1$$

$$\frac{d}{d\theta}\arctan{\theta}=\frac{1}{\sec^2\left(\arctan{\theta}\right)}$$

Now use that identity tan2θ+1=sec2θ.

Chain rule will be useful for inverses, prod./quotient rules will be useful for others.

3. Nov 2, 2005

### CarlB

My latest favorite method of remembering the trig derivatives is this:

$$\frac{d}{dx}e^{ix} = i e^{ix} = e^{ix + \pi/2}$$

From this, by taking real and imaginary parts you get:

$$\frac{d}{dx}\sin(x) = \sin(x+\pi/2)$$

$$\frac{d}{dx}\cos(x) = \cos(x+\pi/2)$$

The derivatives of the trig functions are changes to the phase. You can use trig rules to get the usual forms:

$$\sin(x+\pi/2) = \sin(\pi/2)\cos(x)+ \cos(\pi/2)\sin(x) = \cos(x)$$

$$\cos(x+\pi/2) = \cos(\pi/2)\cos(x)-\sin(\pi/2)\sin(x) = -\sin(x)$$

And always remember, the exponential function is your buddy.

Carl

Last edited: Nov 2, 2005
4. Nov 2, 2005

### Icebreaker

What a coincidence, I have an adcal midterm tomorrow too; not much to trig identities except memorization, unless you want to go from the limit definition each time...