Trigonometry

Trigonometry

1. The Unit Circle

   • terminal side of angle θ
     • segment between the origin and the point
   • coterminal angles
     • two angles with the same terminal side
     • if we add any integer multiple of 360 to an angle θ
       • obtain an angle co-terminal to θ1
         • θ1 +360=θ2
   • π/3
     • 5π/3
       • 4π/3
         • 2π/3
     • cos x is 1/2 but differ positive or negatives depending on quadrant
   • cos x is √3/2 but differ positive or negative depending on quadrant
     • π/6
       • 5π/6
         • 7π/6
           • 11π/6
   • radians and degrees increase counterclockwise
     • the 4 ends circle
       • top π/2
         • left side π
           • bottom 3π/2
             • right side 2π
       • top would be y = 1 since high up
         • left side x=-1 since negative side
           • bottom would be y=-1 since lowest point
             • right would be x=1 since on the positive side
       • all of the coordinates are joined with a 0 for the missing coordinate
   • Radians
     • 2π radians = 360 degrees
       • full rotation
       • π radians = 180 degrees
         • π/2 radians= 90 degrees
           • right angle
         • straight line
     • circumference = 2πr
       • number of arcs= 2π
     • convert radians into degrees
       • multiply the radians by 360/2π
     • convert degrees into radians
       • multiply the angle measure by 2π/360
     • arc length formula
       • s=r|θ|
   • cos x is √2/2
     • same for the sin y
       • just differs positive or negative depending on quadrant
         • π/4
           • 3π/4
             • 5π/4
               • 7π/4

2. Periodic Functions

   • a function f is periodic if its values repeat on regular intervals
     • f is periodic if there exists some constant c
       • f(t+c) =f(t)
       • the period
         • the smallest value |c|
       • the midline
         • the horizontal line midway between the function's minimum and maximum values
           • y=(fmax+fmin)/2
       • the amplitude
         • distance between the function's maximum and the midline
           • or minimum and the midline

3. trigonometric functions

   • sin(θ)
     • vertical coordinate
   • cos(θ)
     • horizontal coordinate
       • θ= 270 degrees
         • θ=3π/2 radians
           • cos(3π/2)=0
             • sin(3π/2)=-1
       • θ=360 degrees
         • θ=2π radians
         • cos(2π)=1
           • sin(2π)=0
   • (x, y) = (cos(θ), sin(θ))
     • θ=90 degrees
       • θ=π/2 radians
       • cos(π/2)=0
         • sin(π/2)=1
     • θ=180 degrees
       • θ=π radians
         • cos(π)=-1
           • sin(π)=0
   • when the terminal side of angle θ is in Quadrant I
     • both x and y are positive
   • when the terminal side of angle θ is in Quadrant II
     • x negative, y positive
     • when the terminal side of angle θ is in Quadrant III
       • both x and y negative
       • when the terminal side of angle θ is in Quadrant IV
         • x positive, y negative
   • tan(θ)=sin(θ)/cos(θ)
   • cot(θ)=1/tan(θ)
   • sec(θ)=1/cos(θ)
   • csc(θ)=1/sin(θ)

4. angles of measure 2π radians and 0 radians specify the same point (1, 0)

   • cos(2π)=cos(0)=1
     • sin(2π)=sin(0)=0
       • since any angle θ and θ + 2π specify the same point on the unit circle
         • the sine and cosine values of θ and θ + 2π are the same
           • the period of the sine and cosine functions is 2π
             • sin(θ)=sin(θ + 2π)
               • cos(θ)=cos(θ + 2π)
                 • both periods = 2π raadians
                   • s(θ)=sin(θ)
                   • c(θ)=cos(θ)
   • the difference
     • between y=cos(θ) and y=sin(θ)
       • shift y=sin(θ) to the left π/2 units and you get the cos(θ) graph
         • or shift cos(θ) graph to the right π/2 units and you get the sin(θ) graph

5. identities

   • one identity sin(θ) =sin(π-θ)
     • focus on the θ-values between 0 and π/2
       • the graphs are the same because of the symmetry of the sine function between 0 and π
   • we can use sine and cosine to find the coordinates of a point on the circumference of any circle
     • construct a right triangle using terminal side of angle θ and the horizontal and vertical components of the point p
       • this right triangle has a hypotenuse of length 1 unit since this is radius
         • a horizontal component of length cos(θ)
           • vertical component of length sin(θ)
             • similar triangles ratio of side-lengths is constant
               • since similar we can obtain the following equations
                 • cos(θ)/1 = x/r
                   • x=rcos(θ)
                 • sin(θ)/1 = y/r
                   • y=rsin(θ)

6. Pythagorean Identity

   • algebraic equation of a unit circle centered at the origin is x^2+y^2=1
     • any point (x, y) on the circumference of a unit circle centered at the origin satisfies the equation
       • (x, y) = (cos(θ), sin(θ))
         • sin^2(θ) + cos^2(θ) = 1
           • example
             • if sin(a)=1/3 and π/2 < a < π
               • we choose the negative square root since cosine is negative in quadrant II
   • by dividing both sides by cos^2(θ) we can find another identity
     • and this one by sin^2(θ)

7. Using geometry for trigonometric functions

   • since the angle is 30 it has to be a 30/60/90 triangle
     • to find the lengths of the sides the triangle has to be mirrored
       • the two halves make the triangle an equilateral triangle
         • if we cut one of the sides in half, each part will be 1/2 of a unit
           • then pythagorean theorem to find a
             • sin(30)=1/2
               • cos(30)=√3/2

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