Mastering Trig Derivatives for Your Calc Midterm

[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

 

[amcavoy]

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

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.

 

[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

 

[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...