Sine Function
Geometric Definition
The sine of a real number t is the ycoordinate (height) of the point P in the following diagram, where t is the length of the arc shown.
sin t = ycoordinate 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:
where
 The baseline (midpoint of vertical oscillation) is situated 2 units below the xaxis
 A = amplitude (the height of each peak above the baseline) = 2
 C = vertical offset = ycoordinate 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 yaxis 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 xcoordinate of the point P in the following diagram, where t is the length of the arc shown.
cos t = xcoordinate of the point P
sin t = ycoordinate 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:
where
 The baseline (midpoint of vertical oscillation) is situated 2 units below the xaxis
 A = amplitude (the height of each peak above the baseline) = 2
 C = vertical offset = ycoordinate 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 yaxis 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
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:
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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 = sec^{2} x 

d
dx 
tan u = sec^{2}u 
du
dx 

d
dx 
cot x =  csc^{2}x 

d
dx 
cot u =  csc^{2}u 
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
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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 


sec^{2}x dx 
= 
tan x + C 


Because 
d
dx 
tan x = sec^{2}x 

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