Geometry and Trigonometry - Trigonometric Identities and Equations

The Unit Circle: The unit circle is a circle with a radius of $1$ centered at the origin $(0,0)$ on the Cartesian plane. For any point $P(x, y)$ on the circumference that forms an angle $\theta$ with the positive x-axis, the coordinates are defined as $x = \cos(\theta)$ and $y = \sin(\theta)$. Visually, this creates a right-angled triangle where the hypotenuse is the radius ($1$), the adjacent side is $x$, and the opposite side is $y$. This relationship ensures that trigonometric functions are periodic, repeating every $2\pi$ radians.

The Pythagorean Identity: Derived directly from the unit circle and the Pythagorean theorem ($a^2 + b^2 = c^2$), we establish that $\sin^2(\theta) + \cos^2(\theta) = 1$. This identity is crucial for converting between sine and cosine in equations. Visually, no matter where the point $P$ is on the circle, the square of the horizontal distance plus the square of the vertical distance always equals the square of the radius ($1^2$).

Double Angle Identities: These identities express functions of $2\theta$ in terms of $\theta$. For example, $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$. Visually, if you look at the graph of $y = \sin(2x)$, it appears 'compressed' compared to $y = \sin(x)$, having half the period and completing two full oscillations in the same interval ($2\pi$) where the original sine graph completes only one.

Solving Trigonometric Equations: To solve equations such as $\sin(x) = k$, we look for all values of $x$ within a specified domain (like $[0, 2\pi]$). Visually, this corresponds to finding the x-coordinates where the horizontal line $y = k$ intersects the trigonometric graph. Due to symmetry, there are usually two solutions within one period (e.g., in the first and second quadrants for a positive sine value).

Reciprocal and Quotient Identities: The tangent function is defined as the ratio of sine to cosine: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. Reciprocal functions include cosecant ($\csc(\theta) = \frac{1}{\sin(\theta)}$), secant ($\sec(\theta) = \frac{1}{\cos(\theta)}$), and cotangent ($\cot(\theta) = \frac{1}{\tan(\theta)}$). Visually, $\tan(\theta)$ represents the slope of the line passing through the origin and the point $P$ on the unit circle.

Quadratic Trigonometric Equations: Some equations appear in the form $a\sin^2(x) + b\sin(x) + c = 0$. These can be solved using algebraic factoring or the quadratic formula by substituting $u = \sin(x)$. Once $u$ is found, we solve for $x$ by identifying the corresponding angles on the unit circle. Visually, these problems involve finding multiple intersection points across several 'waves' of the trigonometric function graph.

Compound Angle Identities: These formulas describe the sine, cosine, or tangent of the sum or difference of two angles, such as $\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)$. Geometrically, these represent the rotation of a vector by one angle and then further adjusting it by a second angle, helping to find exact values for non-standard angles like $15^{\circ}$ or $75^{\circ}$.