Transcript WARM UP write each fraction as a decimal and as a percent (a) ¾ (b

WARM UP
Find the value of the angle θ in degrees:

1.  
6
5
2.  
6
4
3.  
3
What you’ll learn about
• Trigonometric Functions of Any Angle
– Acute angles
– Obtuse angles
– Positive angles
– Negative angles
… and why
• Trigonometry is a mathematical tool that allows
us to solve real-world problems involving right
triangle relationships…we can now move beyond
acute angles, to consider any angle
Vocabulary
• The terms we will use today are:
– Standard position
– Vertex
– Initial side
– Terminal side
– Positive angle
– Negative angle
– Coterminal angles
– Reference triangle
Initial Side, Terminal Side
Vertex
Positive Angle, Negative Angle
Coterminal Angles
Two angles in an extended angle-measurement
system can have the same initial side and the
same terminal side, yet have different measures.
These angles are called coterminal angles.
In other words, what happens when the positive
angle runs into the negative angle?
Coterminal Angles
Coterminal Angles
• Angles of 90˚, 450˚, and 270˚ are all
coterminal
• Angles of π radians, 3π radians, and 9π radians
are all coterminal
• Angles are coterminal whenever they differ by
an integer multiple of 360 degrees or by an
integer multiple of 2π radians
Example: Finding Coterminal Angles
• Find a positive and a negative angle that
are coterminal with 30˚
–Add 360˚
30˚ + 360˚ = 390˚
–Subtract 360˚
30˚– 360˚ = –330˚
Example: Finding Coterminal Angles
• Find a positive and a negative angle that are
coterminal with 30˚: 390˚ and –330˚
Classwork: Finding Coterminal Angles
a) Find a positive and a negative angle
that are coterminal with –150˚
Sketch the angles
b) Find a positive and a negative angle
that are coterminal with 2π/3 radians
Sketch the angles
Investigating First Quadrant Trigonometry
• Let P(x, y) be any point in the first quadrant
(QI), and let r be the distance from P to the
origin
Investigating First Quadrant Trigonometry
• What is sin θ in terms of x, y and/or r?
• What is cos θ in terms of x, y and/or r?
• What is tan θ in terms of x, y and/or r?
Investigating First Quadrant Trigonometry
• Let θ be the acute angle in standard position
whose terminal side contains the point (3, 5).
Find the six trig ratios of θ.
Investigating First Quadrant Trigonometry
The distance from (3,5) to the origin is 34.
5
sin  
 0.857
34
34
csc 
 1.166
5
3
cos 
 0.514
34
5
tan  
3
34
sec 
 1.944
3
3
cot  
5
Slide 4- 15
Example: Trigonometric Functions of any Angle
Find the six trig functions of 315˚
• Reference triangle for 315˚
Example: Trigonometric Functions of any Angle
Find the six trig functions of 315˚
• Draw an angle of 315˚ in standard position
• Pick a point P on the terminal side and connect it to
the x-axis with a perpendicular segment
• The reference triangle formed is a 45-45-90 special
triangle
• Choose the horizontal and vertical sides of the
reference triangle to be of length 1
• P (x, y) has coordinates (1, –1)
Example: Trigonometric Functions of any Angle
Find the six trig functions of 315˚
Trigonometric Functions of any Angle
Let  be any angle in standard position and let P ( x, y ) be any point on the
terminal side of the angle (except the origin). Let r denote the distance from
P( x, y ) to the origin, i.e., let r  x  y . Then
2
y
r
x
cos  
r
y
tan  
( x  0)
x
sin  
r
y
r
sec  
x
x
cot  
y
csc  
2
( y  0)
( x  0)
( y  0)
Evaluating Trig Functions of an Angle θ
1.
2.
3.
4.
5.
Draw the angle θ in standard position, being careful to
place the terminal side in the correct quadrant.
Without declaring a scale on either axis, label a point P
(other than the origin) on the terminal side of θ.
Draw a perpendicular segment from P to the x-axis,
determining the reference triangle. If this triangle is one
of the triangles whose ratios you know, label the sides
accordingly. If it is not, then you will need to use your
calculator.
Use the sides of the triangle to determine the coordinates
of point P, making them positive or negative according to
the signs of x and y in that particular quadrant.
Use the coordinates of point P and the definitions to
determine the six trig functions.
HOMEWORK
Page 381 # 1 to 12
EXIT TICKET
Define one of the following in your own
words:
•
•
•
•
•
•
Vertex
Initial side
Terminal side
Positive angle
Negative angle
Coterminal angles