Study Guide & Educational Overview
### Essential Trigonometric Formulations:
* **Right Triangle Ratios (SOH CAH TOA)**:
* $\sin(\theta) = \text{Opp}/\text{Hyp}$, $\cos(\theta) = \text{Adj}/\text{Hyp}$, $\tan(\theta) = \text{Opp}/\text{Adj}$.
* Reciprocals: $\sec(\theta) = 1/\cos(\theta)$, $\csc(\theta) = 1/\sin(\theta)$, $\cot(\theta) = 1/\tan(\theta)$.
* **Pythagorean Identities**:
* $\sin^2(\theta) + \cos^2(\theta) = 1$
* $1 + \tan^2(\theta) = \sec^2(\theta)$
* $1 + \cot^2(\theta) = \csc^2(\theta)$
* **Unit Circle & Special Values**:
* $\sin(30^\circ) = \cos(60^\circ) = 1/2$; $\sin(45^\circ) = \cos(45^\circ) = \sqrt{2}/2$; $\tan(45^\circ) = 1$.
* Full circle = $2\pi$ radians = $360^\circ$; $\pi$ radians = $180^\circ$; $\pi/2$ = $90^\circ$.
* **Laws for General Triangles**:
* *Law of Sines*: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$.
* *Law of Cosines*: $c^2 = a^2 + b^2 - 2ab\cos(C)$.
* *Triangle Area*: $\text{Area} = \frac{1}{2}ab\sin(C)$.
* **Right Triangle Ratios (SOH CAH TOA)**:
* $\sin(\theta) = \text{Opp}/\text{Hyp}$, $\cos(\theta) = \text{Adj}/\text{Hyp}$, $\tan(\theta) = \text{Opp}/\text{Adj}$.
* Reciprocals: $\sec(\theta) = 1/\cos(\theta)$, $\csc(\theta) = 1/\sin(\theta)$, $\cot(\theta) = 1/\tan(\theta)$.
* **Pythagorean Identities**:
* $\sin^2(\theta) + \cos^2(\theta) = 1$
* $1 + \tan^2(\theta) = \sec^2(\theta)$
* $1 + \cot^2(\theta) = \csc^2(\theta)$
* **Unit Circle & Special Values**:
* $\sin(30^\circ) = \cos(60^\circ) = 1/2$; $\sin(45^\circ) = \cos(45^\circ) = \sqrt{2}/2$; $\tan(45^\circ) = 1$.
* Full circle = $2\pi$ radians = $360^\circ$; $\pi$ radians = $180^\circ$; $\pi/2$ = $90^\circ$.
* **Laws for General Triangles**:
* *Law of Sines*: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$.
* *Law of Cosines*: $c^2 = a^2 + b^2 - 2ab\cos(C)$.
* *Triangle Area*: $\text{Area} = \frac{1}{2}ab\sin(C)$.
Frequently Asked Questions
"All Students Take Calculus": Quadrant I (All positive), Quadrant II (Sine positive), Quadrant III (Tangent positive), Quadrant IV (Cosine positive).
Because tangent is sine divided by cosine; at 90°, cosine equals zero, resulting in division by zero and a vertical asymptote.