Trigonometry - Angles, Degrees and Radian Measure

Angle Measurement and Sense of Rotation: An angle is determined by the amount of rotation of a revolving ray from its initial position to its terminal position. Counter-clockwise rotation is considered positive, while clockwise rotation is negative. Visually, imagine a ray starting on the positive x-axis; rotating it upwards into the first quadrant creates a positive angle, whereas rotating it downwards into the fourth quadrant creates a negative angle.

Sexagesimal System (Degree Measure): In this system, a complete revolution is divided into $360$ equal parts called degrees ($^\circ$). Each degree is further subdivided into $60$ minutes ($'$), and each minute into $60$ seconds ($''$). This can be visualized as a protractor or a clock face where each small graduation represents a specific fractional part of a degree.

Circular System (Radian Measure): A radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. Unlike degrees, radians are real numbers and are more natural for calculus. Visually, if you take the radius of a circle and 'bend' it along the circumference, the angle formed by that arc is exactly $1$ radian ($1^c$).

Relationship between Degrees and Radians: Since a full circle is $360^\circ$ or $2\pi$ radians, the fundamental conversion is $\pi \text{ radians} = 180^\circ$. To convert degrees to radians, multiply by $\frac{\pi}{180}$. To convert radians to degrees, multiply by $\frac{180}{\pi}$. Note that $1$ radian is approximately $57^\circ 16' 22''$.

Arc Length and Sector Area: In a circle of radius $r$, an angle $\theta$ (measured in radians) at the center intercepts an arc of length $s = r\theta$. The region enclosed by two radii and this arc is a sector, with an area calculated as $A = \frac{1}{2}r^2\theta$. Visually, as the angle $\theta$ increases, both the 'crust' length (arc) and the 'size of the slice' (area) increase proportionally.

Trigonometric Ratios on the Unit Circle: A unit circle has a radius of $1$ and is centered at $(0,0)$. For any point $(x, y)$ on the circle at an angle $\theta$ from the positive x-axis, we define $\cos \theta = x$ and $\sin \theta = y$. This extends trigonometry beyond right-angled triangles to any angle in the coordinate plane.

The Quadrant System (ASTC Rule): The coordinate plane is divided into four quadrants. The signs of trigonometric ratios depend on where the terminal side lies: In Quadrant I (All positive), Quadrant II (Sine and Cosec positive), Quadrant III (Tan and Cot positive), and Quadrant IV (Cos and Sec positive). This is visually represented by a cross $(+)$ where each section labels which functions are 'positive' there.