Lesson Solving Trig Functions

Source code of '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(2a) = 2*sin(a)cos(a)}}}
{{{sin(2a) = sin(a + a)}}}
{{{sin(a + a) = sin(a)cos(a) + cos(a)sin(a)}}}
{{{sin(2a) = 2*sin(a)cos(a)}}}
Prove That: {{{cos(2a) = cos(a)^2 - sin(a)^2}}}
{{{cos(2a) = cos(a + a)}}}
{{{cos(a + a) = cos(a)cos(a) - sin(a)sin(a)}}}
{{{cos(2a) = cos(a)^2 - sin(a)^2}}}
*Remember: {{{k}}} is defined as any integer
Solving Sine:
{{{sin(x) - sin(2x) = 0}}}
{{{sin(x) = sin(2x)}}}
{{{x = 2x + 2(pi)k}}}
{{{-x = 2(pi)k}}}
{{{x = -2(pi)k}}}
and
{{{x = (pi) - 2x + 2(pi)k}}}
{{{3x = (pi) + 2(pi)k}}}
{{{x = (pi)/3 + (2(pi)k)/3}}}
{{{graph( 600, 200, -10, 10, -3, 3, sin(x), sin(2x) ) }}}
Solving Absolute Valued Sines:
|sin(x)| = -1
Before you go on, think about this idea. If values from {{{sin(x)}}} are absolute, will a negative number result? No, there are no solutions.
{{{graph( 600, 200, -10, 10, -3, 3, sqrt(sin(x)^2), -1 ) }}}
|sin(x)| = 0
|sin(x)| = sin(0 or {{{pi}}})
{{{x = 0 + 2(pi)k = 2(pi)k}}}
and
{{{x = (pi) + 2(pi)k}}}
and
{{{x = -0 - 2(pi)k = -2(pi)k}}}
and
{{{x = -(pi) - 2(pi)k}}}
To Make The Answer Simple:
x = +-{{{pi}}} + 2{{{pi}}}k
and
x = 2{{{pi}}}k
{{{graph( 600, 200, -10, 10, -3, 3, sqrt(sin(x)^2), 0 ) }}}
Cosines are quite the same:
{{{-2 + cos(x) = -1}}}
{{{cos(x) = 1}}}
{{{cos(x) = cos(0)}}}
{{{x = 2(pi)k}}}
{{{graph( 600, 200, -10, 10, -3, 3, cos(x), 1 ) }}}
Tangent is easy as well:
{{{(sin(x))/(cos(x)) = 0}}}
{{{tan(x) = 0}}}
{{{x = (pi)k}}}
{{{graph( 600, 200, -10, 10, -3, 3, sin(x)/cos(x), 0 ) }}}
or
{{{(sin(x))/(cos(x)) = 0}}}
{{{sin(x) = 0}}}
{{{sin(x) = sin(0)}}}
{{{x = (pi)k}}}
{{{graph( 600, 200, -10, 10, -3, 3, sin(x), 0 ) }}}
Differences That Will Affect Your Solutions:
{{{a*sin(bx + c) + 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( 600, 200, -10, 10, -3, 3, sin(x) ) }}}
NEGATIVE 'a'
{{{graph( 600, 200, -10, 10, -3, 3, -1*sin(x) ) }}}
Lets See What Amplitude Is On A Graph:
Red Line: {{{sin(x)}}}
Green Line: {{{3*sin(x)}}}
Blue Line: {{{-2*sin(x)}}}
{{{graph( 600, 200, -10, 10, -3, 3, sin(x), 3*sin(x), -2*sin(x) ) }}}
Period:
Period (the wavelength of the wave) is described exactly as: {{{(2(pi))/b}}}
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(3x)}}}
{{{(2(pi))/3}}}
{{{graph( 600, 200, -10, 10, -3, 3, sin(3x) ) }}}
{{{sin((1/2)x)}}}
{{{(2(pi))/(1/2)}}}
{{{4(pi)}}}
{{{graph( 600, 200, -10, 10, -3, 3, sin((1/2)x) ) }}}
Let Us Look At A Negative Value For 'b'
{{{y = sin(-x)}}}
{{{graph( 600, 200, -10, 10, -3, 3, sin(-x) ) }}}
Horizontal Shift:
The Horiztonal Shift is the movement along the x-axis: {{{-c}}} units
Example:
{{{sin(x + 1)}}}
{{{-c = -1}}} One unit to the left.
{{{graph( 600, 200, -10, 10, -3, 3, sin(x + 1) ) }}}
More Examples:
Red: {{{sin(x - 4)}}} shifts 4 units right
Green: {{{sin(x + 0.5)}}} shifts 0.5 units left
Blue: {{{sin(x - 1.5)}}} shifts 1.5 units right
{{{graph( 600, 200, -10, 10, -3, 3, sin(x - 4), sin(x + 0.5), sin(x - 1.5) ) }}}
Vertical Shifting: (easiest one)
'd' tells you the units shifted vertically up (positive) or down (negative): {{{d}}} units
{{{sin(x) + 3}}} three units up
{{{graph( 600, 200, -10, 10, -2, 4, sin(x) + 3 ) }}}
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
{{{graph( 600, 200, -10, 10, -3, 3, sin(x) + 1, sin(x) - 2, sin(x) - 0.5 ) }}}