Advanced Function

CHAPTER 5 - Trigonometric function

5.1 Graphs of Sine, Cosine, and Tangent Functions

here's an interesting song! haha:) it's quite entertaining! Well, that is the primary trigonometry identities: [SOH,CAH,TOA]
Sin x = Opposite / Hypotenuse
Cos x = Adjacent / Hypotenuse
Tan x = Opposite / Adjacent

Basic graphs of trigonometric functions are:

Sine Graph

Cosine Graph

Tangent Graph

Graphs y = sin x, y = cos x, y = tan x are periodic.

Original Graph	y = sin x	y = cos x	y = tan x
+ vertical translation v Moves upward/downwards by c units	y = sin x + c	y = cos x + c
+ amplitude v Amplitude increases/decreases by a factor	y = a sin x	y = a cos x	No amplitude because there is no max/min value
+ phase shift v Shifts by d units to the left/right	y = sin (x-d)	y = cos (x-d)
+ period (k-value) v Number of cycles in one period	y = sin kx	y = cos kx	π

5.2 Graphs of Reciprocal Trigonometric Functions

Domain = {x Є R | x ≠ kx } , whereby k is an integer.

Range = { y > 1 or y < -1 }

Period = 2π

Vertical Asymptotes = x = kx , whereby k is an integer

Domain = {x Є R | x ≠ (2kx+1)(π/2) } , whereby k is an integer.

Range = { y > 1 or y < -1 }

Period = 2π

Vertical Asymptotes = x = (2kx+1)(π/2) ‘ whereby k is an integer

Domain = {x Є R | x ≠ kπ } , whereby k is an integer.

Range = { -1< y < 1 }

Period = π

Vertical Asymptotes

5.3 Sinusoidal Functions of the form f(x) = a sin [ k(x-d) ] + c & f(x) = a cos [ k(x-d) ] + c

f(x) = a sin [ k(x-d) ] + c
f(x) = a cos [ k(x-d) ] + c

a = amplitude
k-value = 2 / Period
d = phase shift to left / right
c = vertical translation upwards / downwards

EG:

y = sin x

y = 5 sin x ( where An =5)

y = sin x + 1

= x = kx , whereby k is an integer

5.4 Solve Trigonometric Equations

1.Trigonometric equations can be solved in two ways:

algebraically by hand
graphically with technology

2. There are often multiple solutions. So, be sure to find all solutions that lie in the domain of interest.
3. Quadratic trigonometric ewuations can usually be solved by factoring.

Here are some interesting videos on solving trigonometric equations:) It's very beneficial..hope you guys gain the best of it! Have fun watching!:)

Solving Algebraically