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Math 112.
Elementary Functions.
Chapter 4. Trigonometric functions.
4.1 Angles and radian measure.
An angle is formed by two rays that have a common endpoint. One ray is called
the initial side and the other the terminal side; common point is a vertex.
An angle in standard position if:
Its vertex is at the origin of a rectangular coordinate system and
Its initial side lies along the positive x-axis.
Positive angles are generated by counterclockwise rotation and negative angles
are generated by clockwise rotation.
An angle is called quadrantal angle if its terminal side lies on the x- or y-axis.
Angles can be measured in degrees or radians.
Certain angles have special names. An acute angle measures less than 900, right
angle measures 900, an obtuse angle measures more than 900, and a straight
angle measures 1800.
One radian is the measure of the central angle of a circle that intercepts an arc
equal in length to the radius of the circle. The radian measure of any central angle
s
r
of a circle is the length of the intercepted arc divided by the circle’s radius.   .
In one complete rotation 3600 is the amount of rotation of a ray back into itself. In
radians?
The circumference of a circle of radius r is 2 r ,  
2 r
 2 radians.
r
Using the basic relationship π radians = 1800,
1. To convert degrees to radians, multiply degrees by
 radians
.
1800
1800
2. To convert radians to degrees, multiply radians by
.
 radians
Thinking in radians means determining what part of a complete revolution or how
many full revolutions will produce an angle whose radian measure is known.
Two angles with the same initial side and terminal sides but possibly different
rotations are called coterminal angles.
Increasing or decreasing the degree measure of an angle in standard position by
an integer multiple of 3600 results in a coterminal angle. Thus, an angle of θ0 is
coterminal with angles of   3600 k , where k is an integer.
Increasing or decreasing the radian measure of an angle by an integer multiple of
2π results in a coterminal angle. Thus, an angle of θ radians is coterminal with
angles of θ±2πk, where k is an integer.
The length of a circular arc. Let r be the radius of a circle and θ the nonnegative
radian measure of a central angle of the circle. The length of the arc intercepted
by the central angle is s  r .
Definitions of Linear and Angular speed. If a point is in motion on a circle of
s
t
radius r through an angle of θ radians in time t, then its linear speed is v  ,

t
where s is the arc length given by s  r , and its angular speed is   .
The linear speed, ν, of a point a distance r from the center of rotation is given by
  r ,
where ω is the angular speed in radians per unit of time.
4.2 Trigonometric function: The unit circle.
A unit circle is a circle of radius 1, with its center at the origin of a rectangular
coordinate system. The equation of this unit circle is x 2  y 2  1.
In a unit circle, the radian measure of the central angle is equal to the length of
the intercepted arc.
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The trigonometric functions have names that are words, rather than single letters
such as f, g, etc.
If t is a real number and P = (x , y) is point on the unit circle that corresponds to t,
then
sin t  y
cos t  x
y
tan t  , x  0
x
1
,y0
y
1
sec t  , x  0
x
x
cot t  , y  0
y
csc t 
Because this definition expresses function values in terms of coordinates of a
point on a unit circle, the trigonometric function are sometimes called the circular
functions.
The domain of the sine function and the cosine function is the set of all real
numbers. The range of these functions is the set of all real numbers from -1 to 1,
inclusive.
Trigonometric functions at

2
4
2

2
cos 
4
2
sin
tan

4

1
csc
sec
cot

4

4

4

4
 2
 2
1
Even and odd trigonometric functions.
The cosine and secant functions are even
cos  t   cos t
sec  t   sec t
The rest of the trigonometric functions are odd.
sin  t   t
tan  t   t
csc  t   t
cot  t   t
Fundamental identities:
Reciprocal identities:
1
csc t
1
cos t 
sec t
1
tan t 
cot t
sin t 
1
sin t
1
sec t 
cos t
1
cot t 
tan t
csc t 
Quotient identities:
sin t
cos t
cos t
cot t 
sin t
tan t 
Pythagorean Identities:
sin 2 t  cos 2 t  1
1  tan 2 t  sec 2 t
1  cot 2 t  csc 2 t
Definition of a periodic function.
A function f is periodic if there exists a positive number p such that f  t  p   f t 
for all t in the domain of f. The smallest positive number p for which f is periodic
is called the period of f.
Periodic properties of the Sine and Cosine Functions.
sin  t  2   sin t and cos  t  2   cos t
The sine and cosine functions are periodic functions and have period 2π.
Periodic properties of the Tangent and Cotangent Functions.
tan  t     tan t and cot  t     cot t
The tangent and cotangent functions are periodic functions and have period π.
Repetitive behavior of the sine and cosine, and tangent functions.
For any integer n and real number t,
sin  t  2 n   sin t , cos  t  2 n   cos t , tan  t   n   tan t
4.3 Right triangle trigonometry.
Word trigonometry means measurement of triangles.
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Right triangle definitions of trigonometric functions:
The six trigonometric functions of the acute angle θ are defined as follows:
length of side opposite angle 
length of hypotenuse
length of side adjacant to angle 
cos  
length of hypotenuse
length of side opposite angle 
tan  
length of side adjacant to angle 
sin  
length of hypotenuse
length of side opposite angle 
length of hypotenuse
sec  
length of side adjacant to angle 
length of side adjacant to angle 
cot  
length of side opposite angle 
csc  
The trigonometric function value of θ depend only on the size of the angle θ and
not on the size of the triangle.
Trigonometric functions of special angles:



θ
300 
450 
600 
6
sin 
1
2
cos
3
2
3
3
tan 
4
3
2
2
2
2
3
2
1
2
1
3
Cofunction identities:
The value of a trigonometric function of θ is equal to the cofunction of the
complement of θ. Cofunctions of complementary angles are equal.
sin   cos  900   
cos   sin  900   
tan   cot  900   
cot   tan  900   
sec   csc  900   
csc   sec  900   
If θ is in radians, replace 900 with

.
2
Application.
Many applications of right trigonometry involve the angle made with an
imaginary horizontal line.
An angle formed by the horizontal line and line of sight to an object that is above
the horizontal line is called the angle of elevation. The angle formed by the
horizontal line and line of sight to an object that is below the horizontal line is
called the angle of depression.
4.4 Trigonometric functions of any angle.
Let θ be any angle in standard position and let P ( x , y ) be a point on the terminal
side of θ. If r  x 2  y 2 is the distance from (0, 0) to (x, y), the six trigonometric
functions of θ are defined by the following ratios:
y
r
x
cos  
r
y
tan   , x  0
x
sin  
r
,y0
y
r
sec   , x  0
x
x
cot   , y  0
y
csc  
Reference angle. Let θ be a nonacute angle in standard position that lies in a
quadrant. Its reference angle is the positive acute angle θ̒ formed by the terminal
side of θ and the x-axis.
4.5 Graphs of sine and cosine functions.
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y
10
-10
-5
5
10
x
5
Blue y  sin x
Red y  cos x
-5
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-10
The standard form is
y  A sin B( x  C )
y  A cos B( x  C )
A - amplitude;
2
- period;
B
C - phase shift;

 2  
C , C   B   - appropriate interval on which to sketch one complete period.



4.6 Graphs of other trigonometric functions.
y
10
-10
-5
5
10
x
5
10
x
5
Blue y  tan x
Red y  cot x
-5
y
-10
10
-10
-5
5
Blue y  csc x
Red y  sec x
-5
-10
Standard form
y  A tan B( x  C )
y  A cot B( x  C )


Appropriate interval for tan is    C ,  C  .
2B
2B


Appropriate interval for cot is  C ,  C  .
 B

Standard form

y  A csc B( x  C )
y  A sec B( x  C )
2
Appropriate interval for csc and sec is C ,  C 
 B

4.7 Inverse Trigonometric functions.
y  sin 1 x
sin  sin 1 x   x for x in  1,1
  
sin 1  sin x   x for x in   , 
 2 2
y  cos1 x
cos  cos 1 x   x for x in  1,1
cos1  cos x   x for x in  0,  
y  tan 1 x
tan  tan 1 x   x for every real x
  
tan 1  tan x   x for x in   , 
 2 2
4.8 Applications of trigonometric functions.
Solving a right triangle means finding the missing length of the sides and
measurements of angles.
Simple harmonic motion.
An object that moves on a coordinate axis is in simple harmonic motion if its
distance from the origin, d, at time t is given by either
d  a cos t or d  a sin t .
The motion has amplitude a , the maximum displacement of the object from its
rest position. The period of the motion is
2

, where ω > 0. The period gives the
time it takes for the motion to go through one complete cycle.
Frequency of the object in simple harmonic motion.
An object in simple harmonic motion given by
d  a cos t or d  a sin t
has frequency f given by
f 

,  0 .
2
f 
1
.
period
Equivalently,