Transcript Trigonometric Functions

Trigonometric
Functions
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Chapter 2
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Chapter 2
Overview
 Draw angles in the Cartesian plane.
 Define trigonometric functions as ratios of x and
y coordinates and distances in the Cartesian
plane.
 Evaluate trigonometric functions for nonacute
angles.
 Determine ranges for trigonometric functions
and signs for trigonometric functions in each
quadrant.
 Derive and use basic trigonometric identities.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Chapter 2
Objectives
Skills Objectives
 Plot angles in
standard position.
 Identify coterminal
angles.
 Graph common
angles.
Conceptual Objectives
 Relate the x and y
coordinates to the legs of a
right triangle.
 Derive the distance formula
from the Pythagorean
Theorem.
 Connect angles with
quadrants.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Section 2.1
Angles in the Cartesian Plane
An angle is said to be in
standard position if its
initial side is along the
positive x-axis and its
vertex is at the origin.
We say that an angle lies in the quadrant
in which its terminal side lies.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Angles in Standard Position
Sketching a 210º angle in the
standard position yields this
graph.
•The initial side lies on the x-axis.
•The positive angle indicates
counterclockwise rotation.
•180º represents a straight angle
and the additional 30º yields a
210 º angle.
•The terminal side lies in
quadrant III.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Sketching Angles in Standard Positions
Two angles in standard position with the same
terminal side are called coterminal angles. For
example, -40º and 320º are coterminal angles.
Moving 40º in clockwise direction brings the
terminal side to the same position as moving 320º
in the counter-clockwise direction.
Such angles may also be reached by going the
same direction, such as 90º and 450º. 450º is
reached by moving counterclockwise through the
full 360º circle, then continuing another 90 º.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Coterminal angles
If you graph angles x = 30o and y = - 330o in standard position,
these angles will have the same terminal side. See figure below
Coterminal angles Ac to angle A may be obtained by adding or subtracting
k*360 degrees or k* (2π).
Hence Ac = A + k*360o if A is given in degrees.
Or Ac = A + k*(2π) if A is given in radians;
where k is any negative or positive integer.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Coterminal Angles
Determine the smallest possible measure of
these angles:
 580º
 Solution: Subtract 360º to find the correct
angle of 220º.
 -400º
 Solution: Add 360º to get -40º. Add 360º
again to get the correct angle of 320º.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Your Turn:
Measuring of Coterminal Angles
 Standard Position: An angle is in standard position if its vertex is
located at the origin and one ray is on the positive x-axis. The ray
on the x-axis is called the initial side and the other ray is called the
terminal side. If the terminal side of an angle lies "on" the axes
(such as 0º, 90º, 180º, 270º, 360º ), it is called a quadrantal
angle. The angle shown at the right is referred to as a Quadrant II
angle since its terminal side lies in Quadrant II.
 If Θ is an angle in standard position, and P is any
point (other than the origin) on the terminal side of Θ,
then we associate 3 numbers with the point P.
 x: x-coordinate of the point P
 y: y-coordinate of the point P
 r : distance of the point from the origin
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Common Angles in Standard Position
 The common angles with their exact values
for their Cartesian coordinates are shown
on this graph.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Common Angles in Standard Position
Skills Objectives
Conceptual Objectives
 Calculate trigonometric function  Define trigonometric functions in
the Cartesian plane.
values for acute angles.
 Calculate trigonometric function  Extend right triangle definitions
of trigonometric functions for
values for nonacute angles.
acute angles to definitions of
 Calculate trigonometric function
trigonometric functions for all
values for quadrantal angles.
angles in the Cartesian plane.
 Understand why some
trigonometric functions are
undefined for quadrantal angles.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Section 2.2
Definition 2 of Trigonometric Functions:
Cartesian Plane
 Line up a right triangle with a perpendicular segment
connecting the point (x, y) to the x-axis.
 The distance from the origin, (0, 0), to the point (x, y) is
now:
r  ( x  0) 2  ( y  0) 2  x 2  y 2
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
The Cartesian Plane
 All of the trigonometric functions are defined by
the values of the three sides of a right triangle.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Trigonometric Functions
 For this angle x = 2 and y = 5.
 The distance from the origin is
29 .
y
 sinθ = r =
5
29
2
x
 cosθ =
=
29
r
 tanθ = y = 5
x
2
 The remainder are calculated
from these three values.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Calculating Trigonometric Function Values
 Calculate the values of x, y,
and r in the same way.
 r must be positive.
 For this graph x = -1, y = -3,
and
r=
10 .
 Click for answers!
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Your Turn : Calculating Trigonometric
Functions for Nonacute Angles

Calculate the values of x, y, and r in
the same way.

r must be positive.

For this graph x = -1, y = -3, and
r=
10 .
sinθ =
3
y
=
r
10

cosθ =
x
r

tanθ =
y
=3
x

=
1
10
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Your Turn : Calculating Trigonometric
Functions for Nonacute Angles
The table below summarizes the trigonometric
function values for common quadrantal angles:
0°, 90 °, 180 °, 270 °, and 360 °.
Θ
SINΘ
COSΘ
TANΘ
COTΘ
SECΘ
CSCΘ
0°
0
1
0
U
1
U
90°
1
0
U
0
U
1
180°
0
-1
0
U
-1
U
270°
-1
0
U
0
U
-1
360°
0
1
0
U
1
U
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Quadrantal Values
Skills Objectives
Conceptual Objectives
 Determine algebraic signs of
 Determine the reference
trigonometric functions for all
angle of a nonacute
four quadrants.
angle.
 Determine values for
 Evaluate trigonometric
trigonometric functions for
functions exactly for
quadrantal angles.
common angles.
 Approximate trigonometric  Determine ranges for
trigonometric functions.
functions of nonacute
angles.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Section 2.3
Trigonometric Functions of Nonacute Angles
Sin Θ = y/r
Cos Θ = x/r
Tan Θ = y/x
Csc Θ = r/y , y≠ 0
Sec Θ = r/x , x ≠ 0
Cot Θ = x/y , y ≠ 0
POSITIVE
All
Students
Take
Calculus
y
°
r
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Algebraic Signs of Trigonometric
Functions
If cosθ = -3/5 and the terminal side of the
angle lies in quadrant III, find sinθ.
cosθ = -3/5 means that the x value is
negative, so x = -3 and r = 5.
Now we know that (-3)2 + y2 = 52.
y2 = 25 – 9 = 16, so y = ±4.
Since the angle is in quadrant III, y = -4.
sinθ = y/r = -4/5.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Using the Algebraic Sign of a Trigonometric
Function
 The values of the trigonometric functions for
angles along the axes are undefined for some
angles. For example, along the positive y-axis,
the value of x is zero, making the value of the
tangent undefined.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Values of Quadrantal Trigonometric
Functions
A reference triangle is formed by "dropping" a perpendicular from
the terminal ray of a standard position angle to the xaxis. Remember, it must be drawn to the x-axis.
Reference triangles are used to find trigonometric values for their
standard position angles. They are of particular importance for
standard position angles whose terminal sides reside in quadrants
II, III and IV. A reference triangle contains a reference angle.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Reference Triangle
Skills Objectives
 Learn the reciprocal
identities.
 Learn the quotient
identities.
 Learn the Pythagorean
identities.
 Use the basic identities to
simplify expressions.
Conceptual Objectives
 Understand that
trigonometric reciprocal
identities are not always
defined.
 Understand that quotient
identities are not always
defined.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Section 2.4
Basic Trigonometric Identities
Since sinθ = y/r and cscθ = r/y, these two
trigonometric functions are reciprocals of
one another. Therefore, if y ≠ 0, then cscθ
is defined.
Similarly, cosθ = x/r and secθ = r/x(defined if
x ≠ 0) are reciprocal functions as are
tanθ = y/x (defined if x ≠ 0) and cotθ = x/y
(defined if y ≠ 0) .
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Reciprocal Identities
Since tanθ = sinθ /cosθ and
cotθ = cosθ /sinθ, these two trigonometric
functions are called quotient identities.
Therefore, if cosθ ≠ 0, then tanθ is defined
and if sinθ ≠ 0, then cotθ is defined.
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Quotient Indentities
When studying the unit circle, it was observed that a point on the unit
circle (the vertex of the right triangle)
can be represented by the coordinates (cos Θ, sin Θ ).
Since the legs of the right triangle in the unit circle have the values of
cos Θ and sin Θ, the Pythagorean Theorem can be used to obtain …. .
Pythagorean Identities
Variations
Sin Θ 2 + Cos Θ 2 = 1
Sin Θ 2 = 1 - Cos Θ 2
Cos Θ 2 = 1 - Sin Θ 2
Tan Θ 2 + 1 = Sec Θ 2
Tan Θ 2 = Sec Θ 2 - 1
1 + Cot Θ 2 = Csc Θ 2
Cot Θ 2 = Csc Θ 2 - 1
Trigonometry by Cynthia Y. Young, © 2007 John Wiley and Sons. All rights reserved.
Pythagorean Identities