[Math] Using unit circle to explain $\cos(0) = 1$ and $\sin(90) = 1$
algebra-precalculustrigonometry
We have been taught $\cos(0) = 1$ and $\sin(90) = 1$.
But, how do I visualize these angles on the unit circle?
Best Answer
Suppose you have an angle $\theta$ in the unit circle. Then, the functions $\cos\theta$ and $\sin\theta$ represent the $x$ and $y$ components respectively. See the image below.
Using complex numbers and exponential form perhaps help (at least algebraically) to digest these trigonometric addition formulas:
All we have to know is $\cos a+i\cdot\sin a=e^{ai}$ for any $a\in\Bbb R$, and that $i^2=-1$, and that $e^{x+y}=e^x\cdot e^y$ for any $x,y\in\Bbb C$.
Then calculate both sides of $e^{(a+b)i}=e^{ai}e^{bi}$.
If you prefer, instead, you can use the matrices of rotation:
$$R_a:=\pmatrix{\cos a&-\sin a\\ \sin a &\cos a}$$
and use matrix multiplication to verify the identities, knowing that
$$R_{a+b}=R_a\cdot R_b \ .$$
In your $345$ triangle, you have $\sin x = \frac 35$ and $\cos x = \frac 45$. This gives $x \approx 36.87^\circ$. Can you see where the first two come from the diagram? The $90^\circ$ offset between sine and cosine comes from the fact that we measure horizontally for the cosine and vertically for the sine. While it is true that $\sin x = \cos (90^\circ -x)$ the $90^\circ$ offset is better expressed $\sin x = \cos x - 90^\circ$. This version has $x$ increasing in the same direction on both sides of the equation.
2) opposite is the side opposite the angle. In your diagram, the $3$ side is opposite angle $x$, the $4$ side is adjacent, and $5$ is the hypotenuse. As I said, this gives $\sin x=\frac 35 = 0.6$ It has no units. The $3$ and $5$ have units of length (say, inches) but when you divide them the quotient has no units. The trig functions result in unitless numbers. The $0.6$ gives you one way to construct the angle, though the fact that $\tan x = \frac 34$ is easier to see. You start going east $4$ units, then north $3$ units, and you have constructed the angle.
Best Answer
Suppose you have an angle $\theta$ in the unit circle. Then, the functions $\cos\theta$ and $\sin\theta$ represent the $x$ and $y$ components respectively. See the image below.
Plugging the values $0$ and $90$ you will get it.