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2.4 Trigonometric Functions of General Angles
So far we have been talking about trig functions of acute angles.
We employ the rectangular coordinate system for trig functions of all angles (acute
or not). We place the angle in standard position (vertex at the origin, initial side
on positive x-axis). We’ll let the point (a,b) be on the terminal side of the angle.
y- axis
3
(-2,2)
2
Opposite side
H
-4
-3
yp
ot
e
nu
se 1
Adjacent side
-2
θ
x- axis
0
-1
0
-1
1
origin
(0,0)
2
3
4
-2
-3
-4
We can form a right triangle from any point
(a,b) and the origin (0,0). In the example
above, (a,b) = (-2,2).
The adjacent side = a = -2
Using the actual value of the x-coordinate, a,
instead of the distance identifies the
quadrant of this angle.
The opposite side = b = 2
The hypotenuse, c, is the distance from
(a,b) to the origin
c=
2
2
2
2
(a − 0) + (b − 0) = a + b =
(−2)2 + (2)2 = 8 = 2 2
Trig
Fctn
Definition
Exact Value
sin(θ)
b/c=opp/hyp
b/c=
cos(θ)
a/c=adj/hyp
a/c= − 2 = − 2
2
2
=
2
2 2
2 2
2
tan(θ)
b/a=opp/adj
b/a= 1/-1 = -1
csc(θ)
c/b=hype/opp
c/b= 2 2 = 2
sec(θ)
c/a=hyp/adj
c/a=
cot(θ)
a/b=adj/opp
a/b= -1/1 = -1
2
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2 2
=− 2
−2
Finding the Exact Value of the Six Trig Functions of Quadrantal Angles
3
α= π/2rad = 90°
2
Example 2 on p.154
a)
θ = 0rad = 0°
θ = πrad = 180°
c)
b) θ = π/2rad = 90°
1
d) θ = 3π/2rad = 270°
β= πrad = 180°
-4
a)
y
-3
-2
-1
0° is on the positive x-axis, so choose a
point on the positive x-axis. (a,b)=(1,0)
β0
φ
α
-1
c = square root of (a2 + b2)=square root
of (12 + 02) = 1.
δ= 0rad = 0°
0
1
2
3
δ
-2
Φ= 3π/2rad = 270°
-3
-4
Trig
Funct.
Def.
Exact Value for
θ = 0rad = 0°
[use (a,b)=(1,0)]
Exact Value for
θ = π/2rad = 90°
[use (a,b)=(0,1)]
Exact Value for
θ = πrad = 180°
[use (a,b)=(-1,0)]
Exact Value for
θ = 3π/2rad = 270°
[use (a,b)=(0,-1)]
sin(θ)
b/c=
opp/hyp
b/c=0/1 = 0
b/c=1/1 = 1
b/c=0/1 = 0
b/c=-1/1 = 1
cos(θ)
a/c=
adj/hyp
a/c= 1/1 = 1
a/c= 0/1 = 0
a/c= -1/1 = -1
a/c= 0/1 = 0
tan(θ)
b/a=
opp/adj
b/a= 0/1 = 0
b/a= 1/0
undefined
b/a= 0/-1 = 0
b/a= -1/0
undefined
csc(θ)
c/b=
hyp/opp
c/b=1/0
undefined
c/b=1/1=1
c/b=1/0
undefined
c/b=1/-1= -1
sec(θ)
c/a=
hyp/adj
c/a=1/1=1
c/a=1/0
undefined
c/a=1/-1= -1
c/a=1/0 undefined
cot(θ)
a/b=
adj/opp
a/b= 1/0
undefined
a/b= 0/1 = 0
a/b= =1/0
undefined
a/b= 0/1 = 1
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4
Coterminal Angles
If θ is an angle measured in degrees, then θ ±360°(k) , where k is any integer, is an
angle coterminal with θ.
If θ is an angle measured in radians, then θ ± 2πk , where k is any integer, is an
angle coterminal with θ.
Since coterminal angles are basically equal, all their trigonometry functions are
equal.
Example 3 on p.156
Find the exact value of each of the following:
a)
sin 390°
Reduce 390° to an angle less than 360° by subtracting the highest multiple
of 360° that is less than 390°.
That would be 360x1 = 360
sin(390°) = sin(390° - 360°) = sin(30°) = ½
c)
tan 9π/4
Reduce 9π/4 to an angle less than 2π by subtracting the highest multiple of
2π that is less than 9π/4.
2π = 8π/4
tan 9π/4 = tan (8π/4 + π/4) = tan π/4 = 1
d)
sec(-7π/4)
Change -7π/4 to an angle between 0 and 2π by adding a multiple of 2π.
-7π/4 + 8π/4 = π/4
sec(-7π/4) = sec(-7π/4 + 8π/4) = sec(π/4) = square root of 2.
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Signs of Trigonometric Functions
Quadrant
sin, csc
cos, sec
tan, cot
I
+
+
+
II
+
-
-
III
-
-
+
IV
-
+
-
3
S
II
(-,+)
α= π/2rad = 90°
Sin, csc >0
1
All trig functions >0
cos,sec,tan,cot <0
β= πrad = 180°
-4
-3
δ= 0rad = 0°
0
-2
-1
A
I
(+,+)
2
0
1
2
3
4
-1
III
(-,-)
-2
Φ= 3π/2rad = 270°
Tan,cot > 0
-3
IV
(+,-)
Cos,sec > 0
sin, csc, tan,cot <0
sin, cos, sec, csc <0
-4
T
C
ASTC = “All Surviving Trig take Calculus”
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2.5 Finding the Value of the Trig Functions Using a
Point on the Unit Circle
A unit circle is a circle with its center at the origin (0,0) and
a radius of 1. The equation for this circle is x2+y2 = 1.
Example 1 on p. 165-166
Find the value of sin t, cos t, tan t, csc t, sec t, and cot t
for a point P = (a,b) =  − 1 3 
on a unit circle.
 ,

 2 2 

 cos t = a/1 = a= -½
 −1 3 

( a, b) =  ,

2
2


(0,1)
trad
r= 1
b
sin t = b/1 = b=
tan t = b/a =
θ
3
2
3
3 −1
=
=− 3
2 −1
2
cot t = a/b = −1 3 = −1 =− 3
2 2 3 3
sec t = 1/a = 1 / -½ = -2
a
csc t = 1/b = 1
r
Now let’s look at a
circle with center (0,0) (a, b) (0,r)
and radius r. The
equation for this circle
b
θ
is x2+y2 = r2.
Notice for any point
a
(a,b) on the circle,
a2+b2 = r2.
So r = a 2 + b 2
trad
3 2 2 3
= =
2
3 3
cos t = a/r = a
sin t = b/r = b
tan t = b/a
cot t = a/b
sec t = r/a
csc t = r/b
Now you try #15 on p.172
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Domain and Range of the Trigonometric Functions
( a, b )
(0,1)
r
b
trad
θ
a
Notice that for every right
triangle formed inside a unit
circle with one corner at the
center and another corner on
the circle, the hypotenuse is the
radius, r.
The hypotenuse is always the
longest side of a right triangle.
Therefore |a| ≤r, and
|b| ≤ r.
This mains that the trigonometric functions that are ratios with the hypotenuse
as the denominator are always ≤ 1 and ≥ -1.
This gives us the following
ranges for sin and cos:
The domain of sin and cos is the set of all
sin (θ) = opp/hyp
real numbers. Any number can be used
-1 ≤ sin(θ) ≤ 1
as θ.
sin (θ) = adj/hyp
θ > 360°or 2πrad is just an angle that has
-1 ≤ cos(θ) ≤ 1
made more than 1 full revolution around
the circle.
Reciprocally:
θ < 0 is just an angle that has rotated
csc(θ) ≥ 1 or csc(θ) ≤ -1
clockwise around the circle.
sec(θ) ≥ 1 or sec(θ) ≤ -1
Since csc(θ) is the reciprocal of sin, that is, 1/ sin(θ), then csc(θ) is undefined
when sin(θ) = 0. This occurs when b=0 (on the x-axis). So the domain of
csc(θ) is all real numbers except for integral (integer valued) multples of π.
The domain of sin and cos is the set of all real numbers. Any number can be
used as θ.
θ > 360°or 2πrad is just an angle that has made more than 1 full revolution
around the circle.
θ < 0 is just an angle that has rotated clockwise around the circle.
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Domain and Range of Tan and Cot
(a,b) =(0,1)
π/2
What about tan and cot?
tan(θ) = opp/adj = b/a
If a = 0, then tan(θ) is undefined.
Therefore the domain of tan does not include angles at points
where a=0. Those are all odd multiples of π/2
(i.e. π/2, 3π/2, 5π/2, etc..)
The range of tan(θ) is all real numbers, because when b is
large and a is small, tan(θ) can increase infinitely; and when b
is small and a is large, tan(θ) can decrease infinitely.
cot(θ) = adj/opp = a/b
If b = 0, then cot(θ) is undefined.
Therefore the domain of cot does not include angles at points
where b=0. Those are all integral (integer valued) multiples of
π. (i.e. 0, π, 2π, 3π, 4π, etc..)
Similarly to tan, the range of cot is all real numbers.
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Period of Trig Functions
Review of Coterminal Angles:
If θ is an angle measured in degrees, then θ ±360°(k) , where k is any
integer, is an angle coterminal with θ.
If θ is an angle measured in radians, then θ ± 2πk , where k is any
integer, is an angle coterminal with θ.
Since coterminal angles are basically equal, all their trigonometry
functions are equal.
Therefore
sin(θ + 2π) = sin(θ)
and cos(θ + 2π) = cos(θ)
Functions that exhibit this type of behavior are called periodic
functions.
A function f is periodic if there is a positive number p such that,
whenever θ is in the domain of f, so is θ+p, and f(θ+p) = f(θ).
Let’s revisit our “ASTC” graph:
II
(-,+)
α= π/2rad = 90°
I
(+,+)
1
(a,b)
Sin, csc >0
cos,sec,tan,cot <0
All trig functions >0
π+θ
β= πrad = 180°
-1
III
(-,-)
θ
0
0
(-a,-b)
-1
Φ= 3π/2rad = 270°
Tan,cot > 0
sin, cos, sec, csc <0
δ= 0rad = 0°
1
IV
(+,-)
Cos,sec > 0
sin, csc, tan,cot <0
Going from 0 to 2π, sin(θ)
starts out with the value 0,
then rises to 1 at π/2, then
goes back to 0 at π. At θ>
π, sin(θ) goes from 0 to -1
at 3π/2, then back to 0 at
2π. At this point the sin
values repeat. The period
of the sine function is 2π.
(Likewise for cosine).
-4
Going from 0 to 2π, tan(θ) starts out with the value 0, then rises to
infinity as θ approaches π/2. In Quadrant II, tan(θ) rises from negative
infinity (since the y-coordinates are + and the x-coordinates are -), then
goes back to 0 at π. At θ> π, tan (θ) repeats the same process. The
period of the tangent function is π. (Likewise for cotangent).
tan(θ + π) = tan(θ)
and cot(θ + π) = cot(θ)
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Using Periodic Properties to find exact trig function
values;
Even-Odd Properties
Example 2 on p.171
Find the exact value of
b)
tan 5π/4
Since the period of tan is π, we just need to subtract π from 5π/4
to get a familiar angle for which we know that exact value of
the tan function.
Tan (5π/4) = tan(5π/4 – 4π/4) = tan(π/4).
Recall that π/4 is the same as 45°. At 45°, the point (a,b) on the
unit circle has a=b, so tan(π/4) = a/b = 1.
Now you do #21 on p.173
Even-Odd Properties
A function is even if f(-θ) = f(θ) for all θ in the domain of f.
A function is odd if f(-θ) = - f(θ) for all θ in the domain of f.
Even Trig
Functions
cos(-θ) = cos(θ)
sec(-θ) = sec(θ)
Odd Trig
Functions
sin(-θ) = -sin(θ)
csc(-θ) = -csc(θ)
tan(-θ) = -tan(θ)
cot(-θ) = -cot(θ)
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(a,b)
θ
-θ
(a,-b)
Example 3 on p.172
Find the exact value of
d) tan − 37π 



4 
The tangent function repeats its values for any integral
multiple of π. Therefore, let’s add (since the angle is
37π
negative) a multiple of π to
that will give us a
−
reference angle.
4
36π
= 9π
4
 37π 
 37π 36π 
 π
tan −
+
 = tan −
 = tan − 
4 
 4 
 4
 4
Since tan is an ODD function, we can use the property
that tan(-θ) = -tan(θ).
 π
π 
tan −  = − tan  = −1
 4
4
Now you do #37 and #73 on p.173
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Homework
Ch 2.4 p. 163 #1-117 EOO
Ch. 2.5 #9-84 ETP
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