Download 3_03_DerivativeTrig

Math 1304 Calculus I
3.3 – Derivatives of Trigonometric Functions
Trigonometric Functions
• Overview
Measure
Radians, degrees
Basic functions
sin, cos, tan, csc, sec, cot
Periodicity
Special values at:
0, π/6, π/4, π/3, π/2, π
Sign change
Addition formulas
Derivatives
Angle
• Radians: Measure angle by arc length
around unit circle
θ
Definition of Basic Functions
hypotenuse
opposite
θ
adjacent
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sin() = opposite / hypotenuse
cos() = adjacent / hypotenuse
tan() = opposite / adjacent
csc() = hypotenuse / opposite
sec() = hypotenuse / adjacent
cot() = adjacent / opposite
Sin and Cos Give the Others
sin(  )
tan( ) 
cos( )
1
sec( ) 
cos( )
1
csc( ) 
sin(  )
cos( )
cot( ) 
sin(  )
Sin, Cos, Tan on Unit Circle
tan(θ)
θ
1
sin(θ)
θ
cos(θ)
sin( )    tan( )
Periodicity
sin(   2 )  sin(  )
cos(  2 )  cos( )
tan(  2 )  tan( )
sec(  2 )  sec( )
csc(  2 )  csc( )
cot(  2 )  cot( )
Special Values
sin(0)  0
cos(0)  1
1
sin( /6) 
2
3
cos(/6) 
2
2
cos(/4) 
2
1
cos(/3) 
2
cos(/2)  0
.
2
sin( /4) 
2
3
sin( /3) 
2
sin( /2)  1
.
Basic Inequalities
For
tan(θ)
θ
1
sin(θ)
θ
cos(θ)
0    / 2
sin(  )    tan( )
Proof of Basic Equalities
D
sin(  )  BC  BA  
 sin(  )  
B
tan(θ)
Draw tangent line at B.
It intersects AD at E
θ
1
sin(θ)
O
E
θ
cos(θ)
C
A
  BE  EA
   DE  EA
   tan( )
Special Limit
lim
 0
sin(  )

1
Use Squeezing Theorem
sin(  )    tan( )
 sin(  ) / sin(  )   / sin(  )  tan( ) / sin(  )
 1   / sin(  )  1 / cos( )
 cos( )  sin(  ) /   1
lim cos( )  lim
 0
 0
1  lim
 0
sin(  )

sin(  )

 lim 1
 0
 1  lim
 0
sin(  )

1
Another Special Limit
lim
 0
1  cos( )

0
Addition Formulas
• sin(x+y) = sin(x) cos(y) + cos(x) sin(y)
• cos(x+y) = cos(x) cos(y) – sin(x) sin(y)
Derivative of Sin and Cos
• Use addition formulas (in class)
Derivatives
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If f(x) = sin(x), then f’(x) = cos(x)
If f(x) = cos(x), then f’(x) = - sin(x)
If f(x) = tan(x), then f’(x) = sec2(x)
If f(x) = csc(x), then f’(x) = - csc(x) cot(x)
If f(x) = sec(x), then f’(x) = sec(x) tan(x)
If f(x) = cot(x), then f’(x) = - csc2(x)
A good working set of rules
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Constants: If f(x) = c, then f’(x) = 0
Powers:
If f(x) = xn, then f’(x) = nxn-1
Exponentials: If f(x) = ax, then f’(x) = (ln a) ax
Trigonometric Functions: If f(x) = sin(x), then f’(x)=cos(x)
If f(x) = cos(x), then f’(x) = -sin(x) If f(x)= tan(x), then f’(x) = sec2(x)
If f(x) = csc(x), then f’(x) = -csc(x) cot(x) If f(x)= sec(x), then f’(x) = sec(x)tan(x)
If f(x) = cot(x), then f’(x) = -csc2(x)
Scalar mult: If f(x) = c g(x), then f’(x) = c g’(x)
Sum:
If
f(x) = g(x) + h(x),
then f’(x) = g’(x) + h’(x)
Difference: If
f(x) = g(x) - h(x),
then f’(x) = g’(x) - h’(x)
Multiple sums: derivative of sum is sum of derivatives
Linear combinations: derivative of linear combo is linear combo of derivatives
Product:
If
f(x) = g(x) h(x),
then f’(x) = g’(x) h(x) + g(x)h’(x)
Multiple products: If
F(x) = f(x) g(x) h(x), then F’(x) = f’(x) g(x) h(x) + …
Quotient:
If
f(x) = g(x)/h(x),
then f’(x) = (g’(x) h(x) - g(x)h’(x))/(h(x))2