Math 120 Chapter 3, Section 2 Handout
Verifying Identities
Strategy for Verifying Identities
1) Work on one side of the equation (usually the more complicated side), keeping in
mind the expression on the other side as your goal.
2) Some expressions can be
Math 120 Chapter 1, Section 2 Handout
Radian Measure, Arc Length, and Area
Radians
Radians are another type of measurement of angles. We use a circle of radius, r, to
measure an angle in radians. A central angle is an angle whose vertex is at the center
o
Math 120 Chapter 1, Section 6 Handout
The Fundamental Identity and Reference Angles
The Fundamental Identity
If is any angle or real number, then sin 2 ( ) + cos 2 ( ) = 1 .
Evaluate Trigonometric Functions for a Quadrantal Angle
Draw the angle in standar
Math 120 Chapter 1, Section 4 Handout
The Trigonometric Functions
r
y
x
It is helpful to interpret trigonometric functions in terms of right triangles for certain
kinds of problems.
The Six Trigonometric Functions
If (x, y) is any point other than the ori
Math 120 Chapter 2, Section 3 Handout
Graphs of the Secant and Cosecant Functions
The graphs of y = csc x and y = sec x
We obtain the graphs of cosecant and secant curves by using the reciprocal identities.
csc x = 1/ sin x and sec x = 1/ cos x. y = csc x
Math 120 Chapter 5, Section 1 Handout
The Law of Sines
The Law of Sines and Its Derivation
An oblique triangle is a triangle that does not contain a right angle. An oblique triangle
has either three acute angles or two acute angles and one obtuse angle. T
Math 120 Chapter 4, Section 2 Handout
Basic Sine, Cosine, and Tangent Equations
An identity is satisfied by all values of the variable for which both sides are defined. A
conditional equation is an equation that has at least one solution but isnt an ident
Math 120 Chapter 3, Section 6 Handout
Product and Sum Identities
Product-to-Sum Identities
How do we write the products of sines and/or cosines as sums or differences? We use
the following identities, which are product-to-sum formulas:
1
sin ( A) cos( B )
Math 120 Chapter P, Section 2 Handout
Functions
Relations
A relation is any set of ordered pairs. The set of all first components of the ordered
pairs is called the domain of the relation and the set of all second components is
called the range of the rel
Math 120 Chapter 4, Section 1 Handout
The Inverse Trigonometric Functions
The Inverse Sine Function
For 1 x 1 , y = sin 1 ( x ) or y = arcsin( x ) provided sin ( y ) = x and
y . The
2
2
domain for y = sin 1 ( x ) is [ 1, 1] and its range is , .
2 2
Exa
Math 120 Chapter 2, Section 4 Handout
Graphs of the Tangent and Cotangent Functions
The Graph of y = tan x
Graph y = tan x by listing some points on the graph. Since the period of tangent is ,
graph the function on [0, /2) and complete the graph on the in
Math 120 Chapter 1, Section 5 Handout
Right Triangle Trigonometry
Inverse Sine, Cosine, and Tangent Functions
provided sin ( ) = x and
so quadrants I & IV.
sin 1 ( x ) =
90 90
provided cos( ) = x and
so quadrants I & II.
cos 1 ( x ) =
0 180
provided
Math 120 Chapter 2, Section 1 Handout
The Unit Circle and Graphing
Unit Circle
A unit circle is a circle of radius 1 ( r = 1) with the center of the circle at the origin of
the rectangular coordinate system. The equation of the unit circle is x 2 + y 2 =
Math 120 Chapter 3, Section 5 Handout
Double-Angle and Half-Angle Identities
Double-Angle Formulas
sin ( 2 x ) = 2 sin ( x ) cos( x )
cos( 2 x ) = cos 2 ( x ) sin 2 ( x )
cos( 2 x ) = 2 cos 2 ( x ) 1
cos( 2 x ) = 1 2 sin 2 ( x )
2 tan ( x )
tan ( 2 x ) =
Math 120 Chapter 3, Section 1 Handout
Basic Identities
Reciprocal Identities
1
csc( x )
1
csc( x ) =
sin ( x )
sin ( x ) =
1
sec( x )
1
sec( x ) =
cos( x )
Quotient Identities
tan ( x ) =
sin ( x )
cos( x )
Pythagorean Identities
sin 2 ( x ) + cos 2 ( x )
Math 120 Chapter 2, Section 2 Handout
The General Sine Wave
Period of y = sin ( Bx ) and y = cos( Bx )
The period P of y = sin ( Bx ) and y = cos( Bx ) for B > 0 is given by P =
2
.
B
The General Sine Wave
The graph of y = A sin ( B ( x C ) ) + D or y = A
Math 120 Chapter 1, Section 3 Handout
Angular and Linear Velocity
Angular Velocity
If a point is in motion on a circle through an angle of
angular velocity
is given by =
t
radians in time t, then its
.
Linear Velocity
If a point is in motion on a circle
Math 120 Chapter 4, Section 3 Handout
Multiple-Angle Equations
Equations Involving Multiple Angles
Here are examples of two equations that include multiple angles: tan ( 3x ) = 1 ;
3
x
sin =
2 2 . The period of the trigonometric function plays an importa
Math 120 Chapter 3, Section 4 Handout
Sum and Difference Identities for Sine and Tangent
Sine of a Sum or Difference
sin ( + ) = sin ( ) cos( ) + cos( ) sin ( )
The sine of the sum of two angles equals the sine of the first angle times the cosine of the
s
Math 120 Chapter 5, Section 3 Handout
Area of a Triangle
The Area of an Oblique Triangle
The area of a triangle equals one-half the product of the lengths of two sides times the
1
2
sine of their included angle. This can be expressed by the formulas, Area
Math 120 Chapter 4, Section 4 Handout
Trigonometric Equations of Quadratic Type
Solving Trigonometric Equations
1) Know the solutions to sin ( x ) = a , cos( x ) = a , and tan ( x ) = a .
2) Solve an equation involving multiple angles as if the equation h
Math 120 Chapter 6, Section 1 Handout
Complex Numbers
The Imaginary Unit i
The imaginary unit i is defined as , where , i 3 = i , and i 4 = 1 .
Roots of Negative Number
For any positive real number b, the principal square root of the negative number is
de
Math 120 Chapter 1, Section 1 Handout
Angles and Degree Measure
Angles
A ray is a part of a line that has only one endpoint and extends forever in the opposite
direction.
An angle is formed by two rays that have a common endpoint. One ray is called the
in
Math 120 Chapter P, Section 1 Handout
The Cartesian Coordinate System
Cartesian coordinate system
Many applications of math involve linear equations and their graphs (straight lines).
We then need to begin with a review of the Cartesian coordinate system.