Geometry and Trigonometry - Solving Trigonometric Equations

The Unit Circle: A circle with a radius of $1$ centered at the origin $(0,0)$ where the coordinates of any point $P$ are $(\cos \theta, \sin \theta)$. Visually, as a point moves counter-clockwise from the positive x-axis, the y-coordinate represents the sine value and the x-coordinate represents the cosine value.

The CAST Diagram: A visual tool to identify where trigonometric ratios are positive. In the first quadrant (0 to $90^\circ$ or $\frac{\pi}{2}$), All (A) are positive; in the second, Sine (S) is positive; in the third, Tangent (T) is positive; and in the fourth, Cosine (C) is positive.

Reference Angles: For any angle $\theta$, the reference angle $\theta_{ref}$ is the acute angle formed with the x-axis. To find solutions, we use $\theta_{ref}$ to determine values in other quadrants: Quadrant II is $\pi - \theta_{ref}$, Quadrant III is $\pi + \theta_{ref}$, and Quadrant IV is $2\pi - \theta_{ref}$.

Periodicity and Domain: Trigonometric functions repeat their values over intervals ($2\pi$ for $\sin$ and $\cos$, $\pi$ for $\tan$). When solving equations, always check the specified domain (e.g., $0 \le x \le 2\pi$) to ensure all possible solutions are found.

Solving by Substitution: For equations like $\sin(2x) = k$, we substitute $u = 2x$ and adjust the domain (if $x \in [0, \pi]$, then $u \in [0, 2\pi]$). This visualizes a 'horizontal shrink' where the graph completes cycles more quickly, resulting in more solutions within the original range.

Quadratic Trigonometric Equations: Equations in the form $a\sin^2 x + b\sin x + c = 0$ can be solved by treating $\sin x$ as a variable (e.g., let $u = \sin x$). Visually, this is like finding the x-intercepts of a parabola, but checking if the resulting $u$ values fall within the valid range of $[-1, 1]$.