2 Unit Bridging Course – Day 10
Circular Functions III – The cosine function, identities
and derivatives
Clinton Boys
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The cosine function
The cosine function, abbreviated to cos, is very similar to the
sine function.
In fact, the cos function is exactly the same, except shifted π/2
units to the left.
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The cosine function
The cosine function, abbreviated to cos, is very similar to the
sine function.
In fact, the cos function is exactly the same, except shifted π/2
units to the left.
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Graph of y = cos(x)
Below is the graph of y = cos(x) between x = −4π and x = 4π.
1
−4π − 7π −3π − 5π −2π − 3π
2
2
2
−π
−π
2
−1
π
2
π
3π
2
2π
5π
2
3π
7π
2
4π
The graph continues forever in both directions. Notice the
similarities between cos and sin, as well as the differences.
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Graph of y = cos(x)
Below is the graph of y = cos(x) between x = −4π and x = 4π.
1
−4π − 7π −3π − 5π −2π − 3π
2
2
2
−π
−π
2
−1
π
2
π
3π
2
2π
5π
2
3π
7π
2
4π
The graph continues forever in both directions. Notice the
similarities between cos and sin, as well as the differences.
5 / 31
Graph of y = cos(x)
Below is the graph of y = cos(x) between x = −4π and x = 4π.
1
−4π − 7π −3π − 5π −2π − 3π
2
2
2
−π
−π
2
−1
π
2
π
3π
2
2π
5π
2
3π
7π
2
4π
The graph continues forever in both directions. Notice the
similarities between cos and sin, as well as the differences.
6 / 31
Graph of y = cos(x)
Below is the graph of y = cos(x) between x = −4π and x = 4π.
1
−4π − 7π −3π − 5π −2π − 3π
2
2
2
−π
−π
2
−1
π
2
π
3π
2
2π
5π
2
3π
7π
2
4π
The graph continues forever in both directions. Notice the
similarities between cos and sin, as well as the differences.
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Properties of cosine
cos shares the following properties with sin:
(i) −1 ≤ cos x ≤ 1 for all x.
(ii) cos(x + 2π) = cos x for all x, i.e. cos x is periodic with
period 2π, just like sin x.
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Properties of cosine
Unlike sin, however, cos is not odd:
(iii) cos(−x) = cos(x).
1
−π
2
π
2
−1
y = cos x is symmetric about the y-axis – we say it is an even
function.
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Sketching cosine curves
Practice questions
See if you can sketch the following cosine curves, using the
same ideas we used to sketch sine curves.
(i) y = 2 cos x
(ii) y = cos(2x)
(iii) y = 3 cos(2x).
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Sketching cosine curves
Answers
(i) y = 2 cos x
2
π
2
π
−2
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Sketching cosine curves
Answers
(ii) y = cos(2x)
1
−1
π
4
π
2
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Sketching cosine curves
Answers
(iii) y = 3 cos(2x)
3
π π
4 2
−3
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Identities involving circular functions
Together, sin and cos are called the circular functions.
There are many important identities involving circular functions
which you should remember.
(i) sin2 x + cos2 x = 1 (where sin2 x = (sin x)2 )
(ii) sin(x + y) = sin x cos y + cos x sin y
(iii) cos(x + y) = cos x cos y − sin x sin y
(ii) and (iii) are known as double angle formulas. You can find
plenty more such identities, for example on Wikipedia.
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Identities involving circular functions
Together, sin and cos are called the circular functions.
There are many important identities involving circular functions
which you should remember.
(i) sin2 x + cos2 x = 1 (where sin2 x = (sin x)2 )
(ii) sin(x + y) = sin x cos y + cos x sin y
(iii) cos(x + y) = cos x cos y − sin x sin y
(ii) and (iii) are known as double angle formulas. You can find
plenty more such identities, for example on Wikipedia.
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Identities involving circular functions
Together, sin and cos are called the circular functions.
There are many important identities involving circular functions
which you should remember.
(i) sin2 x + cos2 x = 1 (where sin2 x = (sin x)2 )
(ii) sin(x + y) = sin x cos y + cos x sin y
(iii) cos(x + y) = cos x cos y − sin x sin y
(ii) and (iii) are known as double angle formulas. You can find
plenty more such identities, for example on Wikipedia.
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Derivatives of circular functions
The circular functions, sin and cos, have particularly simple
derivatives.
Derivatives of the circular functions
d
(sin x) = cos x
dx
d
(cos x) = − sin x.
dx
Notice the derivative of cos is negative sin.
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Derivatives of circular functions
The circular functions, sin and cos, have particularly simple
derivatives.
Derivatives of the circular functions
d
(sin x) = cos x
dx
d
(cos x) = − sin x.
dx
Notice the derivative of cos is negative sin.
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Derivatives of circular functions
The circular functions, sin and cos, have particularly simple
derivatives.
Derivatives of the circular functions
d
(sin x) = cos x
dx
d
(cos x) = − sin x.
dx
Notice the derivative of cos is negative sin.
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Derivatives of circular functions
Example
Find the derivative of the function f (x) = 3 sin(2x).
We need to use the chain rule.
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Derivatives of circular functions
Example
Find the derivative of the function f (x) = 3 sin(2x).
We need to use the chain rule.
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Derivatives of circular functions
Example
Find the derivative of the function f (x) = 3 sin(2x).
We need to use the chain rule.
df
dx
= 3 cos(2x) ×
d
(2x)
dx
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Derivatives of circular functions
Example
Find the derivative of the function f (x) = 3 sin(2x).
We need to use the chain rule.
df
dx
d
(2x)
dx
= 3 cos(2x) × 2
= 3 cos(2x) ×
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Derivatives of circular functions
Example
Find the derivative of the function f (x) = 3 sin(2x).
We need to use the chain rule.
df
dx
d
(2x)
dx
= 3 cos(2x) × 2
= 3 cos(2x) ×
= 6 cos(2x).
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Derivatives of circular functions
Example
Find
dy
if y = sin x cos x.
dx
We need to use the product rule. Let u = sin x and v = cos x.
Then
25 / 31
Derivatives of circular functions
Example
Find
dy
if y = sin x cos x.
dx
We need to use the product rule. Let u = sin x and v = cos x.
Then
26 / 31
Derivatives of circular functions
Example
Find
dy
if y = sin x cos x.
dx
We need to use the product rule. Let u = sin x and v = cos x.
Then
dy
dx
= u
dv
du
+v
dx
dx
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Derivatives of circular functions
Example
Find
dy
if y = sin x cos x.
dx
We need to use the product rule. Let u = sin x and v = cos x.
Then
dy
dx
dv
du
+v
dx
dx
= sin x × (− sin x) + cos x × (cos x)
= u
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Derivatives of circular functions
Example
Find
dy
if y = sin x cos x.
dx
We need to use the product rule. Let u = sin x and v = cos x.
Then
dy
dx
dv
du
+v
dx
dx
= sin x × (− sin x) + cos x × (cos x)
= u
= − sin2 x + cos2 x.
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Derivatives of circular functions
Practice questions
Find the derivatives of the following functions:
(i) f (x) = sin2 x
(ii) f (x) = x cos x
(iii) f (x) = sin(x 2 )
sin x
(iv) f (x) =
(usually written tan x).
cos x
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Derivatives of circular functions
Answers to practice questions
(i)
(ii)
(iii)
(iv)
df
dx
df
dx
df
dx
df
dx
= 2 sin x cos x
= −x sin x + cos x
= 2x cos(x 2 )
=
cos2 x+sin2 x
cos2 x
=
1
.
cos2 x
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