Lesson Solving Trig Functions

Here is a neat Trig concept:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
Prove That:



Prove That:



*Remember: is defined as any integer
Solving Sine:





and




Solving Absolute Valued Sines:
|sin(x)| = -1
Before you go on, think about this idea. If values from are absolute, will a negative number result? No, there are no solutions.

|sin(x)| = 0
|sin(x)| = sin(0 or )

and

and

and

To Make The Answer Simple:
x = +- + 2k
and
x = 2k

Cosines are quite the same:





Tangent is easy as well:




or





Differences That Will Affect Your Solutions:
*Generally we use sine for these.
'a' determines the amplitude
'b' determines the period (or wavelenght)
'c' determines the horizontal shift
'd' determines the vertical shift
Amplitude:
Amplitude (the distance from the medium to the crest or trough) is described as: |a|
If 'a' is negative, the wave would be reversed:
POSITIVE 'a'

NEGATIVE 'a'

Lets See What Amplitude Is On A Graph:
Red Line:
Green Line:
Blue Line:

Period:
Period (the wavelength of the wave) is described exactly as:
Frequency tells the closeness of the wave to itself. If the frequency of a sound wave is high, you have a high pitched sound. If low, you have a low pitched sound. The higher 'b' is, the higher the pitch (closer the wave is to itself.)
Example:







Let Us Look At A Negative Value For 'b'


Horizontal Shift:
The Horiztonal Shift is the movement along the x-axis: units
Example:

One unit to the left.

More Examples:
Red: shifts 4 units right
Green: shifts 0.5 units left
Blue: shifts 1.5 units right

Vertical Shifting: (easiest one)
'd' tells you the units shifted vertically up (positive) or down (negative): units
three units up

Examples:
Red: sin(x) + 1 shifts up one unit
Green: sin(x) - 2 shifts two units down
Blue: sin(x) - 0.5 shifts 0.5 units down