6.1 Positive and Negative Angles
- Positive Angle: Angle measured in an anti-clockwise direction from the positive x-axis.
- Negative Angle: Angle measured in a clockwise direction from the positive x-axis.
- Radian Measurement: $\pi \text{ rad} = 180^\circ \implies 1 \text{ rad} = \frac{180^\circ}{\pi} \approx 57.3^\circ$.
6.2 Trigonometric Ratios of Any Angle
Reciprocal Trigonometric Ratios
$$\csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$$
Signs of Trigonometric Functions in Four Quadrants
- Quadrant I ($0^\circ < \theta < 90^\circ$): All ratios ($\sin$, $\cos$, $\tan$) are positive.
- Quadrant II ($90^\circ < \theta < 180^\circ$): Only $\sin$ (and $\csc$) are positive. Reference angle $\alpha = 180^\circ - \theta$.
- Quadrant III ($180^\circ < \theta < 270^\circ$): Only $\tan$ (and $\cot$) are positive. Reference angle $\alpha = \theta - 180^\circ$.
- Quadrant IV ($270^\circ < \theta < 360^\circ$): Only $\cos$ (and $\sec$) are positive. Reference angle $\alpha = 360^\circ - \theta$.
6.3 Graphs of Trigonometric Functions
General Equations
$$y = a \sin (bx) + c, \quad y = a \cos (bx) + c, \quad y = a \tan (bx) + c$$
- Amplitude: $|a|$ (for $\sin$ and $\cos$ graphs). Tangent curves have no maximum/minimum amplitude.
- Period:
- For $\sin$ and $\cos$: $\text{Period} = \frac{360^\circ}{b} = \frac{2\pi}{b}$
- For $\tan$: $\text{Period} = \frac{180^\circ}{b} = \frac{\pi}{b}$
- Vertical Shift ($c$): Shifts the baseline of the graph up or down by $c$ units.
6.4 Basic Identities
Pythagorean Identities
$$\sin^2 \theta + \cos^2 \theta = 1$$
$$1 + \tan^2 \theta = \sec^2 \theta$$
$$1 + \cot^2 \theta = \csc^2 \theta$$
6.5 Addition Formulae and Double Angle Formulae
Addition Formulae
$$\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$$
$$\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B$$
$$\tan (A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$$
Double Angle Formulae
$$\sin 2A = 2 \sin A \cos A$$
$$\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$$
$$\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$$
6.6 Solving Trigonometric Equations
- Simplify the equation using basic, addition, or double angle identities.
- Determine the basic reference angle $\alpha$.
- Identify the correct quadrants based on the sign of the trigonometric ratio.
- Find all possible values of $\theta$ within the specified domain (e.g., $0^\circ \le \theta \le 360^\circ$ or $0 \le \theta \le 2\pi$).