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Here we see a graph of the function y = sin(x), with several tangnet lines to the curve sketched in. |
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Using a technique like that above, numerous slopes of
tangent lines
were then plotted as the red dot values on the graph at the left, along
with the sine function plotted in dark blue.
It appears as though:
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This is confirmed by doing a straight plot of the sine function (red) and cosine function (green) on the same set of axes. |
In an informal way, we have just discovered that the
derivative of the sine function is the cosine function. The textbook
shows formallly that this is true on pages 219-221.
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If you have the Journey Through Calculus CD, load and run MResources/Module 4/Trigonometric Models/Slope-A-Scope for Sine. |
Important Derivative:
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If you have the Journey Through Calculus CD, load and run MResources/Module 4/Trigonometric Models/Slope-A-Scope for Sine. This module will allow you to see another animation. |
In a similar way to the example above, we investigate the
cosine function:
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Here we see a graph of the function y = cos(x), with several tangnet lines to the curve sketched in. |
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Using a technique like that above, numerous slopes of
tangent lines
were then plotted as the red dot values on the graph at the left, along
with the sine function plotted in dark blue.
It appears as though:
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This is verfied by plotting both the cosine curve and a
cosine curve shifted back
radians. In Maple, this would be the command: > plot([cos(x),cos(x+Pi/2)],x=-Pi..2*Pi); We get the exact same results when we plot a cosine
curve with an up-side-down
sine curve:
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In an informal way, we have just discovered that the
derivative of the
cosine function is the additive inverse of the sine function.
Important Derivative:
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To develop a derivative of the tangent, we will use our quotient
rule, and then check our answer with graphical evidence:
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First, we call on one our trigonometric identities. In
one of the ratio identities from trigonometry, we know that:  , so we make this substitution. |
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Here we make use of our quotient rule to take the next step. |
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Now in this step we make use of our new derivatives from this lesson. |
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Simplify algebraically |
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Here we use one of the Pythagorean Identities from
trigonometry, namely,  . |
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And finally, we use another trig identity, one of the
Reciprocal Identities:  . |
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Here we have plotted the tangent function in green and
the secant squared
plotted in yellow.
Does it appear to you that the yellow curve is a correct representation of the derivative of the green curve? First note that the yellow curve is never negative. Is this correct? It makes sense that a squared quantity will always be positive. When is it that the yellow derivative reaches its minimum? This should be an inflection point on the green curve. We see that this is true. |
Important Derivative:
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We have now found derivatives for the sine, cosine, and
tangent functions. What about their reciprocals? Well, we need to use
the reciprocal identities first, and then we will need to use the
quotient rule when we differentiate. We will do only one of these, and
leave the rest to you. We will find the derivative of the cosecant
function:
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We first make use of one the trigonometric reciprocal identities. |
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Here we use the quotient theorem. |
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Note that in the numerator, the derivative of 1 is zero. This form of the answer is just fine, however usually the form below is used. |
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We factor the fraction, |
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And using trig identities, we formulate our final answer. |
Important Derivative:
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We have now found four of the six trigonometric derivatives.
You should
be able to find any of those in this lesson and the remaining two now.
A chart of all six derivatives follows:
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These derivatives need
to be memorized. They will be
coming up often.
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