Fundamental Trigonometric Functions

Fundamental trigonometric functions are essential for understanding angles and triangles. They include sine, cosine, and tangent, which help us analyze relationships in right triangles and model periodic phenomena in real life. These functions are key in various applications.

1. Sine function (sin x)

   • Represents the ratio of the opposite side to the hypotenuse in a right triangle.
   • Defined for all real numbers, with a range of [-1, 1].
   • Key values include sin(0) = 0, sin(π/2) = 1, and sin(π) = 0.
   • Graph is a smooth wave that oscillates between -1 and 1.
   • Period of the sine function is 2π.

2. Cosine function (cos x)

   • Represents the ratio of the adjacent side to the hypotenuse in a right triangle.
   • Also defined for all real numbers, with a range of [-1, 1].
   • Key values include cos(0) = 1, cos(π/2) = 0, and cos(π) = -1.
   • Graph is similar to the sine function but starts at its maximum value.
   • Period of the cosine function is 2π.

3. Tangent function (tan x)

   • Defined as the ratio of sine to cosine: tan(x) = sin(x)/cos(x).
   • Has a range of all real numbers, with vertical asymptotes where cos(x) = 0.
   • Key values include tan(0) = 0, tan(π/4) = 1, and tan(π/2) is undefined.
   • Graph has a repeating pattern with vertical asymptotes at odd multiples of π/2.
   • Period of the tangent function is π.

4. Cosecant function (csc x)

   • Defined as the reciprocal of sine: csc(x) = 1/sin(x).
   • Undefined where sin(x) = 0, leading to vertical asymptotes at integer multiples of π.
   • Range is (-∞, -1] ∪ [1, ∞).
   • Graph has a repeating pattern with vertical asymptotes at odd multiples of π.
   • Period of the cosecant function is 2π.

5. Secant function (sec x)

   • Defined as the reciprocal of cosine: sec(x) = 1/cos(x).
   • Undefined where cos(x) = 0, leading to vertical asymptotes at odd multiples of π/2.
   • Range is (-∞, -1] ∪ [1, ∞).
   • Graph has a repeating pattern with vertical asymptotes at odd multiples of π/2.
   • Period of the secant function is 2π.

6. Cotangent function (cot x)

   • Defined as the ratio of cosine to sine: cot(x) = cos(x)/sin(x).
   • Has a range of all real numbers, with vertical asymptotes where sin(x) = 0.
   • Key values include cot(0) is undefined, cot(π/4) = 1, and cot(π/2) = 0.
   • Graph has a repeating pattern with vertical asymptotes at integer multiples of π.
   • Period of the cotangent function is π.

7. Unit circle and its relationship to trigonometric functions

   • A circle with a radius of 1 centered at the origin of the coordinate plane.
   • Coordinates of points on the circle correspond to (cos θ, sin θ) for an angle θ.
   • Helps visualize the values of sine and cosine for different angles.
   • Provides a way to define trigonometric functions for all real numbers.
   • Key angles (0, π/6, π/4, π/3, π/2, etc.) have specific sine and cosine values.

8. Periodic nature of trigonometric functions

   • Trigonometric functions repeat their values in regular intervals.
   • Sine and cosine functions have a period of 2π, while tangent and cotangent have a period of π.
   • This periodicity allows for the prediction of function values over intervals.
   • Important for solving equations and modeling periodic phenomena.
   • Graphs of these functions exhibit repeating wave patterns.

9. Amplitude, period, and phase shift of trigonometric functions

   • Amplitude refers to the height of the wave from the midline to the peak (for sine and cosine).
   • Period is the length of one complete cycle of the wave.
   • Phase shift is the horizontal shift left or right of the graph.
   • These properties can be altered by coefficients in the function's equation.
   • Understanding these concepts is crucial for graphing and analyzing trigonometric functions.

10. Inverse trigonometric functions

• Functions that reverse the action of the original trigonometric functions.
• Include arcsin, arccos, and arctan, which return angles for given ratios.
• Defined within specific ranges to ensure they are functions (e.g., arcsin is limited to [-π/2, π/2]).
• Useful for solving equations involving trigonometric functions.
• Graphs of inverse functions reflect the original functions across the line y = x.