Derivatives of Trigonometric Functions

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Derivatives of Trigonometric Functions

• Deriving the Derivative of sin(x)
• The Derivative of cos(x)
• Derivatives of the other Trigonometric Functions
• Examples

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The Derivative of sin x

• To find the derivative of sin x, we need to
  recall two limits we discovered in lab.

and
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Using these limit results, we can use the
definition of the derivative to find derivatives
of sin x and cos x.
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Using similar techniques we can show that if y
cos x, then y - sin x.
Using the formulas derived for sin x and cos x,
we can use the quotient rule to find the
derivatives of the other 4 trigonometric
functions
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So, we have
You should be able to derive the derivatives of
the final three trigonometric functions
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Find the 93rd derivative of sin x

• f (x) cos x
• f (x) - sin x
• f (x) - cos x
• f iv (x) sin x
• f v (x) cos x
• So, f (93) (x) f (f (92) (x)) f (sin
  x)
• cos x

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Example
For what values of x does the function above
have a horizontal tangent line?
Where do horizontal tangentlines occur on the
graph of afunction ?
At local max/min points. (humps)
Local max/min points generally occur where the
derivative is equal to zero.
So we need to find the x-values that make the
derivative equal to zero.
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Example(contd)
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Example(contd)
So, now we have
Now if f (x) is to be zero, either sec x or
tan x 1 must equal zero.
Recall the graph of sec x ?
You should note that secant is never zero.
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Example(contd)
And sec x ? 0, sotan x 1 must 0
We have
So, tan x 1. What values of x make this
true?
tan x ?
tan x 1 when x n? ?/4
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Verifying Results
This is the graph of the original function
Note how the horizontal tangent lines occur at
multiples of ?/4.