Lesson Solving Trig Functions
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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 ) }}}