6.1 Trigonometric Ratios in Quadrants I, II, III, and IV
Unit Circle and Signs of Functions
In a unit circle with radius $r = 1$, the coordinates of a point $P(x, y)$ corresponding to an angle $\theta$ measured counterclockwise from the positive x-axis are given by:
$$\sin\theta = y, \quad \cos\theta = x, \quad \tan\theta = \frac{y}{x} \quad (x \neq 0)$$
ASTC Quadrant Rule
- Quadrant I ($0^\circ \le \theta \le 90^\circ$): All positive ($\sin, \cos, \tan > 0$). Reference angle $\alpha = \theta$.
- Quadrant II ($90^\circ < \theta \le 180^\circ$): Sine positive. Reference angle $\alpha = 180^\circ - \theta$.
- Quadrant III ($180^\circ < \theta \le 270^\circ$): Tangent positive. Reference angle $\alpha = \theta - 180^\circ$.
- Quadrant IV ($270^\circ < \theta \le 360^\circ$): Cosine positive. Reference angle $\alpha = 360^\circ - \theta$.
Exact Values of Special Angles
$$\begin{array}{|c|c|c|c|c|c|}
\hline
\theta & 0^\circ & 30^\circ & 45^\circ & 60^\circ & 90^\circ \\
\hline
\sin\theta & 0 & \frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{\sqrt{3}}{2} & 1 \\
\hline
\cos\theta & 1 & \frac{\sqrt{3}}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} & 0 \\
\hline
\tan\theta & 0 & \frac{1}{\sqrt{3}} & 1 & \sqrt{3} & \text{Undefined} \\
\hline
\end{array}$$
6.2 Graphs of Sine, Cosine, and Tangent Functions
Basic Trigonometric Curves ($0^\circ \le x \le 360^\circ$)
- $y = \sin x$: Starts at $(0,0)$, maximum $1$ at $90^\circ$, zero at $180^\circ$, minimum $-1$ at $270^\circ$, zero at $360^\circ$. Amplitude = $1$, Period = $360^\circ$.
- $y = \cos x$: Starts at $(0,1)$, zero at $90^\circ$, minimum $-1$ at $180^\circ$, zero at $270^\circ$, maximum $1$ at $360^\circ$. Amplitude = $1$, Period = $360^\circ$.
- $y = \tan x$: Passes through origin, increases to $+\infty$ at $90^\circ$ and $270^\circ$ (vertical asymptotes). Period = $180^\circ$ (no amplitude).
Transformations of Trigonometric Graphs
For functions of the form $y = a \sin(bx) + c$, $y = a \cos(bx) + c$, or $y = a \tan(bx) + c$:
$$\text{Amplitude} = |a| \quad (\text{for sine and cosine only})$$
$$\text{Period} = \frac{360^\circ}{b} \quad (\text{for sine and cosine}), \quad \text{Period} = \frac{180^\circ}{b} \quad (\text{for tangent})$$
$$c = \text{Vertical shift / baseline position}$$