How is that differentiation of $\cos(\theta)$ possible?

Question

I pondered upon this transformation in a book and have some confusion about it. $$y =\cos(\theta) = \sin\left(\frac\pi2-\theta\right)$$$$dy = d\left(\sin\left(\frac\pi2-\theta\right)\right)=\cos\left(\frac\pi2-\theta\right) \times d(-\theta) = \cos\left(\frac\pi2-\theta\right) \times (-d\theta)$$

$$\frac {dy}{d\theta}=-\cos\left(\frac\pi2-\theta\right)=-\sin(\theta)$$

The main part I don't understand is: $$dy = d\left(\sin\left(\frac\pi2-\theta\right)\right)=\cos\left(\frac\pi2-\theta\right) \times d(-\theta)$$

This is (from what I can tell) an informal way to differentiate $\cos(\theta)$.

This is a very odd way of writing a corollary of the chain rule — FShrike, Apr 18 '22 at 19:13
It assumes that you know the derivative of $\sin$ and it uses the chain rule. — Andrei, Apr 18 '22 at 19:13
This is just the chain rule. If you draw the graphs of $\sin$ and $\cos$ you can see how the derivatives are related. — Ethan Bolker, Apr 18 '22 at 19:14
By the way, you can type \cos and \sin to get $\cos$ and $\sin$ (rather than $cos$ and $sin$, which respectively refer to the products $c \cdot o \cdot s$ and $s \cdot i \cdot n$). We definitely appreciate the effort to MathJax, though! — Brian Tung, Apr 18 '22 at 19:26

score 3 · Accepted Answer · edited Apr 18 '22 at 21:35

3

The chain rule says that if f(y) is a differentiable function of y and y(x) is a differentiable function of x, then $\frac{df}{dx}= \frac{df}{dy}\frac{dy}{dx}$.

In this example, $f(y)= \sin(y)$ and $y(x)= \frac{\pi}{2}- x$. $\frac{df}{dy}= \cos(y)$ and $\frac{dy}{dx}= -1$.

So with $f(x)= \cos(x)= \sin(\pi- x)$ $\frac{df}{dx}= \frac{df}{dy}\frac{dy}{dx}= \cos(y)(-1)= -\cos\left(\frac\pi2- x\right)= -\sin(x)$.

edited Apr 18 '22 at 21:35

hash1

5

answered Apr 18 '22 at 19:37

George Ivey

1,497

Thank you for your thorough explanation. – hash1 Apr 18 '22 at 20:01

score 0 · Answer 2 · answered Apr 18 '22 at 19:18

0

Your questions boils down to the chain rule, a fundamental aspect of elementary differentiation. If you are not familiar with it, check out this post on the intuition of the chain rule.

answered Apr 18 '22 at 19:18

Clyde Kertzer

2,904

How is that differentiation of $\cos(\theta)$ possible?

2 Answers2