Trigonometric functions

Trigonometric functions

1. Unit circle

   • Is a circle with a radius of 1 centered at the origin (0,0) on a Cartesian graph.
   • It is the fundamental tool in trigonometry used to evaluate and define trigonometric functions for any real number or angle.

2. Trigonometric ratios

   • Compare the lengths of the side of a right-angled triangle
   • The three primary ratios are Sin (sin), Cosine (cos) and Tangent (tan).
   • They are defined relative to one of the triangle’s acute angles (θ)

3. Pythagorean Identities

   • The three fundamental Pythagorean identities are foundational equations in trigonometry derived from the Pythagorean theorem and the unit circle

4. Inverse trigonometric functions

   • Are inverse operations of the standard trigonometric functions.
   • While regular trig functions take an angle and output a side ratio, inverse trig functions take a side ratio and output the corresponding angle.

5. Graph of the sine function (y=sin x) Graph of the cosine function (y=cos x)

   • Those graphs are periodic
   • Continuous wave-like curves called sinusoids that oscillate between a minimum value of -1 and a maximum value of 1.

6. Graph of the tangent function (y=tan x)

   • Consists of repeating vertical curves separated by vertical asymptotes. It increases from left to right within each period.

7. Asymptotes

   • A line or curve that a mathematical function’s graph approaches infinitely closely, but never actually touches.

8. Graph of the cotangent function (y=cot x)

   • Consists of a repeating series of downward-sloping curves separated by vertical asymptotes.

9. Graph of the secant function (y=sec x)

   • Consists of a repeating series of open U-shaped and inverted U-shaped curves separated by vertical asymptotes.

10. Graph of the cosecant function (y=csc x)

    • Consists of repeating U-shaped curves separated by vertical lines called asymptotes.

11. Translation of functions (Shifts: vertical/horizontal)

    • A translation moves a function’s graph horizontally or vertically without changing its shape or size.

12. Reflection of functions (Across axes/origin)

    • Involves “flipping” a graph across a specific line or point-
    • Across the X-Axis: The entire output of the function is negated.
    • Across the Y-Axis: Only the input variable inside the function is negated
    • Across the origin: Both the input and output are flipped.

13. Compression and stretching (Vertical/horizontal scaling)

    • Are transformations that alter the size and shape of a function’s graph.
    • Vertical transformations apply to the entire output (the y-values)
    • Horizontal transformations apply directly to the input (the x-value) and generally behave in the opposite way you might expect.

14. Amplitude (Height of oscillations)

    • Is the maximum displacement of an oscillating object or wave from its central, neutral position.
    • It physically represents the “height” of a wave’s crest or the “depth” of its through

15. Period (Length of one complete cycle)

    • The period is the distance along the horizontal axis required for a function to complete one full cycle of its repeating pattern.
    • Once the period is complete, the wave repeats its value exactly.

16. Phase shift (Horizontal displacement)

    • Is the horizontal translation of a periodic function or wave along the x-axis
    • It dictates exactly how far a wave has been shifted left or right from its standard, baseline position.