Trig Functions in Cartesian Plane

The Unit Circle

A circle centered at the origin with radius 1:

$x^2 + y^2 = 1$

For any point $P(x, y)$ on the unit circle at angle $\theta$:

$x = \cos\theta$

$y = \sin\theta$

Definition Using the Unit Circle

Function	Definition
$\sin\theta$	y-coordinate of point on unit circle
$\cos\theta$	x-coordinate of point on unit circle
$\tan\theta$	$\frac{y}{x} = \frac{\sin\theta}{\cos\theta}$
$\csc\theta$	$\frac{1}{y} = \frac{1}{\sin\theta}$
$\sec\theta$	$\frac{1}{x} = \frac{1}{\cos\theta}$
$\cot\theta$	$\frac{x}{y} = \frac{\cos\theta}{\sin\theta}$

Standard Position Angle

An angle in standard position has:

• Vertex at the origin
• Initial side along positive x-axis
• Terminal side determined by rotation

Positive angle: Counter-clockwise rotation Negative angle: Clockwise rotation

Reference Angles

The reference angle $\theta'$ is the acute angle between the terminal side and the x-axis.

Quadrant	Reference Angle
I	$\theta' = \theta$
II	$\theta' = 180° - \theta = \pi - \theta$
III	$\theta' = \theta - 180° = \theta - \pi$
IV	$\theta' = 360° - \theta = 2\pi - \theta$

Signs by Quadrant

Quadrant	$x$ ($\cos$)	$y$ ($\sin$)	$\tan$
I	$+$	$+$	$+$
II	$-$	$+$	$-$
III	$-$	$-$	$+$
IV	$+$	$-$	$-$

Memory Aid: "All Students Take Calculus"

• All functions positive in Quadrant I
• Sine positive in Quadrant II
• Tangent positive in Quadrant III
• Cosine positive in Quadrant IV

Coterminal Angles

Angles with the same terminal side.

$\theta \text{ and } \theta + 360°n \text{ are coterminal (degrees)}$

$\theta \text{ and } \theta + 2\pi n \text{ are coterminal (radians)}$

General Angle Definition

For any angle $\theta$ with terminal point $P(x, y)$ at distance $r$ from origin:

$\sin\theta = \frac{y}{r}$

$\cos\theta = \frac{x}{r}$

$\tan\theta = \frac{y}{x}$

$\csc\theta = \frac{r}{y}$

$\sec\theta = \frac{r}{x}$

$\cot\theta = \frac{x}{y}$

where $r = \sqrt{x^2 + y^2} > 0$

Quadrantal Angles

Angles whose terminal side lies on an axis.

Angle	Point	$\sin$	$\cos$	$\tan$
$0°$	$(1, 0)$	$0$	$1$	$0$
$90°$	$(0, 1)$	$1$	$0$	undef
$180°$	$(-1, 0)$	$0$	$-1$	$0$
$270°$	$(0, -1)$	$-1$	$0$	undef
$360°$	$(1, 0)$	$0$	$1$	$0$

Finding Trig Values from a Point

Given point $P(x, y)$ on terminal side of $\theta$:

1. Find $r = \sqrt{x^2 + y^2}$
2. Apply definitions:
   • $\sin\theta = \frac{y}{r}$
   • $\cos\theta = \frac{x}{r}$
   • $\tan\theta = \frac{y}{x}$

Example

Point $(-3, 4)$ on terminal side:

• $r = \sqrt{9 + 16} = 5$
• $\sin\theta = \frac{4}{5}$
• $\cos\theta = -\frac{3}{5}$
• $\tan\theta = -\frac{4}{3}$

Evaluating Using Reference Angles

1. Find the reference angle $\theta'$
2. Evaluate trig function at $\theta'$
3. Apply appropriate sign based on quadrant

Example: $\sin 210°$

• Quadrant III, reference angle $= 210° - 180° = 30°$
• $\sin 30° = \frac{1}{2}$
• In Q III, sin is negative
• $\sin 210° = -\frac{1}{2}$