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Applied calculus topic summary: trigonometric functions

Tools: Function Evaluator & Grapher | Excel Grapher

Subtopics: Sine Function | Cosine Function | Fundamental Trigonometric Identities | The Other Trigonometric Functions | Derivatives of Trigonometric Functions | Indefinite Integrals of Trigonometric Functions

Sine Function

Geometric Definition
The sine of a real number t is the y-coordinate (height) of the point P in the following diagram, where |t| is the length of the arc shown.


sin t = y-coordinate of the point P

"Bicycle Wheel" Definition
If a wheel of radius 1 unit rolls forward at a speed of 1 unit per second, sin t is the height after t seconds of a marker on the rim of the wheel, starting midway between the top and bottom of the wheel.


   

Graph of the Sine Function


y = sin x

General Sine Function
The "generalized sine function" has the following form:


y = A sin[ω(x - α)] + C
  • A is the amplitude (the height of each peak above the baseline).
  • C is the vertical offset (height of the baseline).
  • P is the period or wavelength (the length of each cycle).
  • ω is the angular frequency, and is given by
            ω= 2π/P or P = 2π/ω.
  • α is the phase shift.

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Examples

Consider the following graph, depicting a shifted and scaled ("general") sine curve:

Question What is the equation of this curve?
Answer Referring to the generalized sine function at the bottom of the column on the left, we see that the equation of this curve is:

    y = A sin[ω(x-α)] + C,
where
  • The baseline (midpoint of vertical oscillation) is situated 2 units below the x-axis
  • A = amplitude (the height of each peak above the baseline) = 2
  • C = vertical offset = y-coordinate of baseline = -2
  • P = period (the length of each cycle, or distance from one peak to the next) = 4
  • ω = angular frequency = 2π/P = 2π/4 = π/2
  • α = phase shift = 1 This is the horizontal distance from the y-axis to the first point where the graph intersects the baseline
Thus, the equation of the above curve is
    y = 2 sin[π/2 (x - 1)] - 2

To test that it works, try it out on the excel grapher (if you have Excel on your computer).

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Cosine Function

Geometric Definition
The cosine of a real number t is the x-coordinate of the point P in the following diagram, where |t| is the length of the arc shown.


cos t = x-coordinate of the point P
sin t = y-coordinate of the point P

Graph of the Cosine Function


y = cos x

General Cosine Function
The "generalized sine function" has the following form:


y = A cos[ω(x - α)] + C
  • A is the amplitude (the height of each peak above the baseline).
  • C is the vertical offset (height of the baseline).
  • P is the period or wavelength (the length of each cycle).
  • ω is the angular frequency, and is given by
            ω= 2π/P or P = 2π/ω.
  • α is the phase shift.

Examples

Consider the following graph -- the same one as above, depicting a shifted and scaled ("general") sine curve:

Question This time, what is the equation of this curve expressed as a general cosine function?
Answer Referring to the generalized cosine function at the bottom of the column on the left, we see that the equation of this curve is:

    y = A cos[ω(x-α)] + C,
where
  • The baseline (midpoint of vertical oscillation) is situated 2 units below the x-axis
  • A = amplitude (the height of each peak above the baseline) = 2
  • C = vertical offset = y-coordinate of baseline = -2
  • P = period (the length of each cycle, or distance from one peak to the next) = 4
  • ω = angular frequency = 2π/P = 2π/4 = π/2
  • α = phase shift = 2 This is different for cosine: the horizontal distance from the y-axis to the top of the first peak
Thus, the equation of the above curve is
    y = 2 cos[π/2 (x - 2)] - 2

To test that it works, try it out on the excel grapher (if you have Excel on your computer).

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Fundamental Trigonometric Identities: Relationships Between Sine and Cosine

The sine and cosine of a number t are related by

    sin2t + cos2t = 1
We can obtain the cosine curve by shifting the sine curve to the left a distance of π/2. Conversely, we can obtain the sine curve from the cosine curve by shifting it π/2 units to the right. These facts can be expressed as
    cos t = sin(t + π/2)
    sin t = cos(t - π/2)

Alternative formulation

We can also obtain the cosine curve by first inverting the sine curve vertically (replace t by -t) and then shifting to the right a distance of π/2. This gives us two alternative formulas (which are easier to remember):

    cos t = sin(π/2 - t)     Cosine is the sine of the complement.
    sin t = cos(π/2 - t) Sine is the cosine of the complement.

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Examples

From the first identity on the left, we get:

sin2x = 1 + cos2x
1 - cos2x
cos2x - 1

From the next-to-last identity on the left, we get:

cos π/3 = sin π/2
sin π/3
sin π/6

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The Other Trigonometric Functions

The ratios and reciprocals of sine and cosine are given their own names:

    Tangent
    tan x =
    sin x

    cos x
    Cotangent:
    cot x = cot x =
    cos x

    sin x
    =
    1

    tan x
    Secant:
    sec x =
    1

    cos x
    Cosecant:
    csc x = csc x =
    1

    sin x

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Derivatives of Trigonometric Functions

The following table summarizes the derivatives of the six trigonometric functions, as well as their chain rule counterparts (that is, the sine, cosine, etc. of a function).

Original Rule Generalized Rule
(Chain Rule)
d

dx
sin x = cos x
d

dx
sin u = cos u
du

dx
d

dx
cos x = - sin x
d

dx
cos u = - sin u
du

dx
d

dx
tan x = sec2 x
d

dx
tan u = sec2u
du

dx
d

dx
cot x = - csc2x
d

dx
cot u = - csc2u
du

dx
d

dx
sec x = sec x tan x
d

dx
sec u = sec u tan u
du

dx
d

dx
csc x
=- csc x cot x
d

dx
csc u
=- csc u cot u
du

dx

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Example

1.
d

dx
x sin x =
1.sin x + x cos x   Product rule
=
sin x + x cos x
2.
d

dx
cos(2x2+1) =
sin(2x2+1)
d

dx
(2x2+1)
=
sin(2x2+1).4x = 4x sin(2x2+1)
3.
d

dx
sec(x3) =
sec(x3) tan(x3)
d

dx
(x3)
=
sec(x3) tan(x3) . 3x2
=
3x2 sec(x3) tan(x3)
5.
d

dx
x cos(x2) =

Use proper graphing calculator format
     

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Indefinite Integrals of Trigonometric Functions
    sin x dx = -cos x + C
    Because
    d

    dx
    -cos x = sin x
    cos x dx = sin x + C
    Because
    d

    dx
    sin x = cos x
    tan x dx = -ln |cos x| + C
    Because
    d

    dx
    -ln |cos x| = tan x
    cot x dx = ln |sin x| + C
    sec x dx = ln |sec x + tan x| + C
    csc x dx = -ln |csc x + cot x| + C
    sec2x dx = tan x + C
    Because
    d

    dx
    tan x = sec2x

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Last Updated: July 2007
Copyright © Stefan Waner

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