Trigonometry: complete lesson (unit circle, sin, cos, tan, formulas)

The unit circle

The unit circle is the circle centered at $O$ with radius $1$ in an orthonormal frame.

To a real number $x$, we associate a point $M$ on the circle such that the arc length from $(1, 0)$ equals $x$ (counter-clockwise).

• $\cos(x)$ = x-coordinate of $M$
• $\sin(x)$ = y-coordinate of $M$
• $\tan(x) = \dfrac{\sin(x)}{\cos(x)}$ (defined when $\cos(x) \neq 0$)

Key values to memorize

$x$	$0$	$\pi/6$	$\pi/4$	$\pi/3$	$\pi/2$
$\cos(x)$	$1$	$\sqrt{3}/2$	$\sqrt{2}/2$	$1/2$	$0$
$\sin(x)$	$0$	$1/2$	$\sqrt{2}/2$	$\sqrt{3}/2$	$1$
$\tan(x)$	$0$	$\sqrt{3}/3$	$1$	$\sqrt{3}$	∞

Memory hint: $\cos$ decreases, $\sin$ increases on $[0, \pi/2]$.

Fundamental properties

Periodicity

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

$\tan$ has period $\pi$: $\tan(x + \pi) = \tan(x)$.

Parity

$\cos(-x) = \cos(x) \quad \text{(even)}$ $\sin(-x) = -\sin(x) \quad \text{(odd)}$

Pythagorean identity

$\cos^2(x) + \sin^2(x) = 1$

Associated angles

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

Addition formulas

$\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)$ $\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)$ $\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)$ $\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)$

Double-angle formulas

$\cos(2a) = \cos^2(a) - \sin^2(a) = 2\cos^2(a) - 1 = 1 - 2\sin^2(a)$ $\sin(2a) = 2\sin(a)\cos(a)$

Solving $\cos(x) = a$ and $\sin(x) = a$

$\cos(x) = a$: solutions $x = \pm \alpha + 2k\pi$ where $\alpha = \arccos(a)$.

$\sin(x) = a$: solutions $x = \alpha + 2k\pi$ or $x = \pi - \alpha + 2k\pi$.

Functions $x \mapsto \cos(x)$ and $x \mapsto \sin(x)$

• Defined on $\mathbb{R}$
• Periodic with period $2\pi$
• Values in $[-1, 1]$
• $\cos$ is even, $\sin$ is odd

Common mistakes

1. Forgetting $+2k\pi$ in equation solutions.
2. Confusing $\cos(a+b)$ with $\cos(a) + \cos(b)$: wrong!
3. Forgetting $\cos^2 + \sin^2 = 1$ to eliminate a variable.
4. Sign errors in $\cos(\pi - x)$ vs $\cos(\pi + x)$.