Comprehensive Trigonometry and Graphing Guide

Special Right Triangles

• 45-45-90 Triangle:     * Angles: $45^{\circ}$, $45^{\circ}$, and $90^{\circ}$.     * Side proportions: The legs are equal to $1$, and the hypotenuse is $\\sqrt{2}$.

• 30-60-90 Triangle:     * Angles: $30^{\circ}$, $60^{\circ}$, and $90^{\circ}$.     * Side proportions: The side opposite the $30^{\circ}$ angle is $1$, the side opposite the $60^{\circ}$ angle is $\\sqrt{3}$, and the hypotenuse is $2$.

The Unit Circle and Quadrants

• Unit Circle Coordinates:     * $0^{\circ}$ or $360^{\circ}$ ($2\\pi$): $(1, 0)$     * $90^{\circ}$ ($\\frac{\\pi}{2}$): $(0, 1)$     * $180^{\circ}$ ($\\pi$): $(-1, 0)$     * $270^{\circ}$ ($\\frac{3\\pi}{2}$): $(0, -1)$

• Quadrant Signs (ASTC Rule):     * Quadrant I (A - All): All trigonometric functions ($\\sin(x)$, $\\cos(x)$, $\\tan(x)$, $\\csc(x)$, $\\sec(x)$, $\\cot(x)$) are positive $(+,+)$.     * Quadrant II (S - Sine): Only Sine and Cosecant are positive $(-,+)$.     * Quadrant III (T - Tangent): Only Tangent and Cotangent are positive $(-,-)$.     * Quadrant IV (C - Cosine): Only Cosine and Secant are positive $(+,-)$.

Trigonometric Identities

• Cofunction Identities:     * $\\sin(90 - \\theta) = \\cos(\\theta)$     * $\\cos(90 - \\theta) = \\sin(\\theta)$     * $\\tan(90 - \\theta) = \\cot(\\theta)$     * $\\csc(90 - \\theta) = \\sec(\\theta)$     * Specific Example: $\\cos(90 - 20) = \\sin(20)$

• Even and Odd Identities:     * $\\sin(-\\theta) = -\\sin(\\theta)$     * $\\cos(-\\theta) = \\cos(\\theta)$     * $\\tan(-\\theta) = -\\tan(\\theta)$     * $\\csc(-\\theta) = -\\csc(\\theta)$

Trigonometric Functions and Graphing Properties

• General Sine/Cosine Form:     * Standard Equation: $y = A \\sin(Bx)$ or $y = A \\cos(Bx)$     * $A$: Amplitude     * $\\text{Period} = \\frac{2\\pi}{B}$

• Secant Function ($y = \\sec(x)$):     * Domain: $x \\neq \\frac{\pi}{2} + \\pi n$     * Range: $(-\infty, -1] \\cup [1, \\infty)$     * Period: $2\\pi$

• Cosecant Function ($y = \\csc(x)$):     * Domain: $x \\neq 0 + \\pi n$     * Range: $(-\infty, -1] \\cup [1, \\infty)$     * Period: $2\\pi$

• Tangent Function ($y = \\tan(x)$):     * Period: $\\pi$

• Cotangent Function ($y = \\cot(x)$):     * Domain: $x \\neq 0 + \\pi n$     * Range: $(-\infty, \\infty)$     * Period: $\\pi$

• Function Transformations Noted:     * $y = -\\cos(x)$     * $y = -\\tan(x)$     * $y = -\\cot(x)$

Additional Topics

• Inverse Trigonometric Functions:     * Included as a topic of study, categorized as "Inverse/Arc Trig Functions".

• Miscellaneous Notes:     * UFCPS9 (notation/reference marker)     * SING     * F-Sing     * Values noted: $0.21-6$, $0.300^{\circ}$, $0.0338$.