Download Lecture 9

Trigonometric Functions
September 22, 2014
In this lecture, we will discuss a type of function which is omitted from
your textbooks, though is common enough in practical applications that it
is worth discussing. These are the trigonometric functions: functions which
relate the ratio of sides of a triangle to the angles of the triangle. These functions appear in many different fields, such as chemistry, engineering, physics,
etc. We will start with a brief review of the three standard trigonometric
functions, and then we will discuss how to compute their derivatives.
Definitions of Trig Functions
Consider a right triangle (a triangle where one of the angles is 90◦ ):
We will use Greek letter θ to denote the angle indicated in the picture, and the
lengths of the sides will be labeled as o (for opposite), a (for adjacent), and
h (for hypotenuse). Using these definitions, we can define three quantities,
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known as the sine, cosine, and tangent of the angle θ, defined as
opposite
hypotenuse
adjacent
cos θ =
hypotenuse
opposite
tan θ =
adjacent
sin θ =
Example 1. Suppose that a = 4, o = 3, and h = 5. Then for the corresponding angle θ, we have
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sin θ =
5
4
cos θ =
5
3
tan θ =
4
Note that this does not tell us what the angle θ is. It only tells us the ratio
of the sides of the triangles in such a triangle.
Alternatively, we could be given the angle θ in a triangle, and we would
like to know the ratio of the sides of a corresponding triangle. Computing
these ratios for any angle requires some pretty sophisticated math, and is too
complicated for this course. The important thing is that this has been done
and is programmed into scientific and graphing calculators. (In the past, if
you needed to know the value of a trig function for a specific angle, you would
look at a chart full of numbers.) There are some special angles for which the
trig functions yield fairly simple values. These are 0◦ , 30◦ , 45◦ , 60◦ , and 90◦ .
The values of the trig functions at these angles is contained in the following
table.
θ
0◦
30◦
sin θ
0
cos θ
1
1
2
√
3
2
tan θ
0
45◦
60◦
√
√
2
2
√
2
2
√1
3
1
90◦
3
2
1
1
2
0
√
3
Undef.
These may seem complicated, but for most other angles, the corresponding
value of the trig functions are complicated sequences of decimals which cannot
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be written as simple fractions like we have done here. Thus, we will tend to
focus only on the angles listed above.
In the definitions above, there is something which you may not have noticed regarding the definitions of the trig functions. From the way we defined
the trig functions, it would seem that if we think of them as functions, their
domains would be between 0◦ and 90◦ . We can, however, extend the definitions so that they are defined for any angle between 0◦ and 360◦ . However,
by using the following definitions, we can extend each of these functions to
any angle we like.

sin θ
if 0◦ ≤ θ ≤ 90◦



sin(180◦ − θ)
if 90◦ ≤ θ ≤ 180◦
sin θ =

− sin(θ − 180◦ ) if 180◦ ≤ θ ≤ 270◦



− sin(360◦ − θ) if 270◦ ≤ θ ≤ 360◦
For angles greater than 360◦ , we simply subtract 360◦ from the given angle
repeatedly until we obtain an angle less than 360◦ , and then apply the above
definitions. Additionally, we can define sin θ for negative angles by
sin θ = − sin(−θ).
By using the values in the table above, combined with the definitions for
larger angles, you can obtain the following graph for the sine function.
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We can do the same thing for cosine by using the following definition:

cos θ
if 0◦ ≤ θ ≤ 90◦



− cos(180◦ − θ) if 90◦ ≤ θ ≤ 180◦
cos θ =

− cos(θ − 180◦ ) if 180◦ ≤ θ ≤ 270◦



cos(360◦ − θ)
if 270◦ ≤ θ ≤ 360◦
For angles larger than 360◦ , we just reduce by 360◦ until we obtain an angle
less than 360◦ . For negative angles, we use
cos θ = cos(−θ).
This should give you the following graph:
To complete our discussion, we define

tan θ



− tan(180◦ − θ)
tan θ =

tan(θ − 180◦ )



− tan(360◦ − θ)
the tangent function by
if
if
if
if
0◦ ≤ θ ≤ 90◦
90◦ ≤ θ ≤ 180◦
180◦ ≤ θ ≤ 270◦
270◦ ≤ θ ≤ 360◦
We treat large (> 360◦ ) in the same way as for sine and cosine, and we treat
negative angles according to the formula
tan θ = − tan(−θ).
This gives you the following graph:
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Note that the vertical lines are not actually there, but a limitation of the
graphing software. For our purposes, they represent asymptotes in the tangent function.
Example 2. Compute the following trigonometric functions using the values
given above.
1. sin(135◦ )
√
Solution: sin(135◦ ) = sin(180◦ − 135◦ ) = sin(45◦ ) = 2/2.
2. cos(240◦ )
Solution: cos(240◦ ) = − cos(240◦ − 180◦ ) = − cos(60◦ ) = 1/2.
3. tan(405◦ )
Solution: tan(405◦ ) = tan(405◦ − 360◦ ) = tan(45◦ ) = 1.
The Unit Circle and Radian Measure
Consider a circle of radius one, and an angle inscribed into the circle, as in
the following diagram.
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We know that the distance around the entire circle (the circumference) for
a circle of radius 1 is just 2π. Thus, the angle in the picture contains some
portion of this distance around the whole circle. This portion of the entire
circumference is known as the radian measure of the angle, and is given by
2π
r=
d,
360◦
where r is the radian measure, and d is the number of degrees in the angle.
Mathematicians and scientists prefer to use radian measure over angles, and
so it is important to know how to use them.
Example 3. Convert the following angles to radians.
1. 45◦
Solution: Using the formula above, we get
45◦
1
2π
π
◦
45
=
2π
·
= 2π · = .
r=
◦
◦
360
360
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2. 240◦
By the formula, we have
240◦
2
2π
4π
◦
240
=
2π
·
= 2π · =
r=
◦
◦
360
360
3
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Example 4. Evaluate the following trig functions.
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1. cos
π 4
Solution: Based on√the previous example, we know that
cos π4 = cos 45◦ = 22 .
π
4
= 45◦ . Thus
2. tan( 4π
)
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Solution: Using the computation from the previous example, we know
that 4π
= 240◦ . Thus
3
4π
1
cos
= cos(240◦ ) = − cos(240◦ − 180◦ ) = − cos(60◦ ) = − .
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Derivatives of Trigonometric Functions
The final thing we would like to do with trig functions is differentiate
them. As always, we will determine their derivatives by using the definition. We begin with sine. Assume f (x) = sin(x). By the definition of the
derivative, we have
f (x + h) − f (x)
h→0
h
sin(x + h) − sin(x)
= lim
.
h→0
h
f 0 (x) = lim
To proceed, we must make use of some identities which are obeyed by trig
functions. These are the content of the following theorem.
Theorem 1. The following identities hold:
sin(a + b) = sin a cos b + cos a sin b
cos(a + b) = cos a cos b − sin a sin b
If we apply the first one to the equation above, we get
sin(x + h) − sin(x)
h→0
h
sin(x) cos(h) + cos(x) sin(h) − sin(x)
= lim
h→0
h
cos(h) − 1
sin(h)
= sin(x) lim
+ cos(x) lim
h→0
h→0
h
h
f 0 (x) = lim
To finish this, we need the following fact:
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Theorem 2. The following limits hold.
cos(h) − 1
lim
=0
h→0
h
sin(h)
lim
= 1.
h→0
h
Applying this to our computation, we finally get
d
(sin(x)) = cos(x).
dx
By performing a similar computation, you can show that
d
(cos(x)) = − sin(x).
dx
Finally, for homework, you will show that
d
1
(tan(x)) =
.
dx
cos2 (x)
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