Comprehensive Guide to Geometry, Coordinate Geometry, and Trigonometry

Lines and Slopes: This subject involves the fundamental properties of linear equations in a coordinate plane.
- Slope Calculation: The slope, often represented by the variable $m$ , indicates the steepness and direction of a line. It is calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
- Finding the y-intercept: The y-intercept represents the point where a line crosses the vertical $y$ -axis. In the standard slope-intercept form $y = mx + b$ , the y-intercept is denoted by the constant $b$ , corresponding to the coordinate $(0, b)$ .
- Parallel Lines: Two lines are considered parallel if they lie in the same plane and never intersect. For two lines to be parallel, their slopes must be exactly equal, such that $m_1 = m_2$ .
- Perpendicular Lines: Lines that intersect at a right angle ( $90^\circ$ ) are perpendicular. The relationship between their slopes is that they are negative reciprocals of one another, expressed as $m_1 = -\frac{1}{m_2}$ , which implies that the product of their slopes equals $-1$ ( $m_1 \times m_2 = -1$ ).

Circles:
- The perimeter of a circle, known as the circumference, is found using the formula $C = 2 \times \pi \times r$ or $C = \pi \times d$ , where $r$ is the radius and $d$ is the diameter.
- The area of a circle is calculated as $A = \pi \times r^2$ .
Triangles:
- The perimeter of a triangle is the sum of its three sides: $P = a + b + c$ .
- The area of a triangle is determined by the formula $A = \frac{1}{2} \times \text{base} \times \text{height}$ , or $A = \frac{1}{2} \times b \times h$ .
Rectangles:
- The perimeter of a rectangle is found by summing all four sides, expressed as $P = 2 \times (l + w)$ , where $l$ is length and $w$ is width.
- The area of a rectangle is calculated by multiplying the length by the width: $A = l \times w$ .
Volume:
- Volume measures the three-dimensional space occupied by an object. For rectangular prisms, the formula is $V = l \times w \times h$ . For cylinders, the volume is defined as the area of the circular base multiplied by the height, or $V = \pi \times r^2 \times h$ .

Application: This theorem is used specifically to solve for missing side lengths in right-angled triangles (triangles containing one $90^\circ$ angle).
The Formula: The relationship between the sides is given by $a^2 + b^2 = c^2$ .
- In this equation, $a$ and $b$ represent the lengths of the two legs that form the right angle.
- The variable $c$ represents the length of the hypotenuse, which is the longest side of the triangle and is opposite the right angle.

Circle Equations: A circle is the set of all points equidistant from a center point $(h, k)$ . The basic equation is $(x - h)^2 + (y - k)^2 = r^2$ , where $r$ is the radius.
Parabolas: Parabolas are U-shaped curves that can open upward, downward, left, or right. The basic standard form of a vertical parabola is $y = ax^2 + bx + c$ . The vertex form is often expressed as $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.

Definitions: Trigonometric ratios define the relationship between the angles and sides of a right triangle.
SOH: Sine is the ratio of the Opposite side to the Hypotenuse, represented as $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ .
CAH: Cosine is the ratio of the Adjacent side to the Hypotenuse, represented as $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ .
TOA: Tangent is the ratio of the Opposite side to the Adjacent side, represented as $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$ .

Conceptual Overview: The unit circle is a circle with a radius of $1$ centered at the origin $(0, 0)$ of the coordinate plane. It is used to find trigonometric values for any angle.
Exact Values for Common Angles:
- $30^\circ$ ( $\frac{\pi}{6}$ radians): $\sin(30^\circ) = \frac{1}{2}$ , $\cos(30^\circ) = \frac{\sqrt{3}}{2}$ .
- $45^\circ$ ( $\frac{\pi}{4}$ radians): $\sin(45^\circ) = \frac{\sqrt{2}}{2}$ , $\cos(45^\circ) = \frac{\sqrt{2}}{2}$ .
- $60^\circ$ ( $\frac{\pi}{3}$ radians): $\sin(60^\circ) = \frac{\sqrt{3}}{2}$ , $\cos(60^\circ) = \frac{1}{2}$ .
- $90^\circ$ ( $\frac{\pi}{2}$ radians): $\sin(90^\circ) = 1$ , $\cos(90^\circ) = 0$ .

Pythagorean Identities: These identities are derived from the Pythagorean Theorem as applied to the unit circle.
Fundamental Identity: A primary identity used in calculations is $\sin^2(x) + \cos^2(x) = 1$ .

Sine and Cosine Curves: These functions produce periodic oscillatory waves.
Amplitude: The amplitude represents the peak height of the wave from the center line. In the function $y = A \sin(x)$ , the amplitude is the absolute value of $A$ ( $|A|$ ).
Period: The period is the horizontal length of one complete cycle of the wave. For standard sine and cosine graphs, the period is $2 \times \pi$ . This can be altered by a coefficient $B$ in the form $\sin(Bx)$ , where the period becomes $\frac{2 \times \pi}{B}$ .
Phase Shifts: A phase shift represents a horizontal displacement of the graph. In the equation $y = \sin(x - C)$ , the value $C$ determines the shift along the $x$ -axis.