Lesson Solving Trig Functions

This Lesson (Solving Trig Functions) was created by by Nate(3500)

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: sin%282a%29+=+2%2Asin%28a%29cos%28a%29

sin%28a+%2B+a%29+=+sin%28a%29cos%28a%29+%2B+cos%28a%29sin%28a%29

Prove That: cos%282a%29+=+cos%28a%29%5E2+-+sin%28a%29%5E2

cos%28a+%2B+a%29+=+cos%28a%29cos%28a%29+-+sin%28a%29sin%28a%29

*Remember:

is defined as any integer
Solving Sine:
sin%28x%29+-+sin%282x%29+=+0

and

x+=+%28pi%29%2F3+%2B+%282%28pi%29k%29%2F3

graph%28+600%2C+200%2C+-10%2C+10%2C+-3%2C+3%2C+sin%28x%29%2C+sin%282x%29+%29+

Solving Absolute Valued Sines:
|sin(x)| = -1
Before you go on, think about this idea. If values from sin%28x%29

are absolute, will a negative number result? No, there are no solutions.
graph%28+600%2C+200%2C+-10%2C+10%2C+-3%2C+3%2C+sqrt%28sin%28x%29%5E2%29%2C+-1+%29+

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

)

and

and

and

To Make The Answer Simple:
x = +-

+ 2

k
and
x = 2

graph%28+600%2C+200%2C+-10%2C+10%2C+-3%2C+3%2C+sqrt%28sin%28x%29%5E2%29%2C+0+%29+

Cosines are quite the same:
-2+%2B+cos%28x%29+=+-1

graph%28+600%2C+200%2C+-10%2C+10%2C+-3%2C+3%2C+cos%28x%29%2C+1+%29+

Tangent is easy as well:
%28sin%28x%29%29%2F%28cos%28x%29%29+=+0

graph%28+600%2C+200%2C+-10%2C+10%2C+-3%2C+3%2C+sin%28x%29%2Fcos%28x%29%2C+0+%29+

graph%28+600%2C+200%2C+-10%2C+10%2C+-3%2C+3%2C+sin%28x%29%2C+0+%29+

Differences That Will Affect Your Solutions:
a%2Asin%28bx+%2B+c%29+%2B+d

*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'
graph%28+600%2C+200%2C+-10%2C+10%2C+-3%2C+3%2C+sin%28x%29+%29+

graph%28+600%2C+200%2C+-10%2C+10%2C+-3%2C+3%2C+sin%28x%29+%29+

NEGATIVE 'a'
graph%28+600%2C+200%2C+-10%2C+10%2C+-3%2C+3%2C+-1%2Asin%28x%29+%29+

Lets See What Amplitude Is On A Graph:
Red Line: sin%28x%29

Green Line: 3%2Asin%28x%29

Blue Line: -2%2Asin%28x%29

Period:
Period (the wavelength of the wave) is described exactly as: %282%28pi%29%29%2Fb

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:
sin%283x%29