Download Trigonometric Functions The Unit Circle

Trigonometric Functions
The Unit Circle
The unit circle is a circle with radius one, centred at the point (0, 0). The
equation of the unit circle is
x2 + y 2 = 1
√
For example the point (
√
3
6
,
)
3
3
is on the unit circle. Why?
For every real positive number t, we can start from the point (1, 0) and
move counter-clockwise around the unit circle for t units. The point we arrive
at is called the terminal point of t. If t is negative, we travel clockwise.
This process gives us an arc on the circle of length t.
Recall that the circumference of the unit circle is 2π.
Example: Find the terminal points for the following numbers. Draw the
unit circle for each.
arc
2π
π
π/2
3π/2
0
3π
−π/2
−π
terminal point
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Example Find the terminal point for t = π/4.
Similarly, we can prove the following table.
arc terminal point
0
(1,
0)
√
3 1
π/6
(√2 ,√2 )
π/4
( 22 ,√22 )
π/3
( 12 , 23 )
π/2
(0, 1)
Important Observation: For any real number t, the terminal point for t
is equal to terminal point for t + 2kπ for any integer k.
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Exercise Use the above table to find the terminal points of the following
numbers.
t = −π/4
t = −5π/6
t = 5π/6
t = 7π/4
t = 29π/6
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Trigonometric Functions
We learned that to every real number, we can assign a point (the terminal
point) on the unit circle. We use the x and y coordinates of this point to
define several functions. Let P (x, y) be the point on the unit circle defined
by t. The trigonometric functions are defined as follows:
1. The function that assigns the value of y to t is called the sin function
sin(t) = y
2. The function that assigns the value of x to t is called the cos function
cos(t) = x
3. The function that assigns the value of
y
x
tan(t) =
y
x
4. The function that assigns the value of
x
y
cot(t) =
x
y
5. The function that assigns the value of
1
y
csc(t) =
1
y
6. The function that assigns the value of
1
x
sec(t) =
1
x
to t is called the tan function
to t is called the cot function
to t is called the csc function
to t is called the sec function
Example: Find the value of all the trig functions for t =
4
π
3
and t = π.
Observe that the functions sec and tan are not defined for those values
of t whose terminal point has the x-coordinate 0. Also, the functions csc
and cot are not defined for those values of t whose terminal point has the
y-coordinate 0.
Use the unit circle to show that the domains of the trig functions are as
follows:
Dtan
Dsin = Dcos = R
π
= Dsec = {t | t 6= + kπ, k ∈ Z}
2
Dcot = Dcsc = {t | t 6= kπZ}
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The table below gives you some of the values of trig functions.
The sign of the trig functions at a given point t depends on the quadrant
in which the terminal point of t lies in. To remember which function is positive in which quadrant, you can travel counter clockwise around the circle
and say ”All Students Take Calculus“.
Example: Determine the sign of the cos(t), sin(t) tan(t) and cot(t) at the
following values for t.
t = π4 ,
t=
2π
,
3
t = − π6 ,
5π
.
6
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In general, we have
sin(−t) = − sin(t)
csc(−t) = − csc(t)
cos(−t) = cos(t)
tan(−t) = − tan(t)
sec(−t) = − sec(t) cot(−t) = − cot(t)
Exercise: Use the unit circle to justify the above equalities.
In general we have the following fundamental properties.
csc t =
1
,
sin t
sec t =
1
,
cos t
cot t =
1
tan t
sin t
cos t
, cot =
cos t
sin t
2
2
sin t + cos t = 1
tan t =
Example: If cos t =
functions at t.
3
5
and t is in the 4th quadrant, find all other trig
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