Download Evaluating trigonometric functions.

Evaluating trigonometric functions.
Evaluating trigonometric functions.
1 / 13
Last time
Using the techniques we developed last time, evaluate:
cos(0)
sin(π)
tan 2π
cos(13π)
(each 2 · π contributes a full revolution.)
Evaluating trigonometric functions.
2 / 13
Last time
Using the techniques we developed last time, evaluate:
cos(0) 1
sin(π)
tan 2π
cos(13π)
(each 2 · π contributes a full revolution.)
Evaluating trigonometric functions.
2 / 13
Last time
Using the techniques we developed last time, evaluate:
cos(0) 1
sin(π) 0
tan 2π
cos(13π)
(each 2 · π contributes a full revolution.)
Evaluating trigonometric functions.
2 / 13
Last time
Using the techniques we developed last time, evaluate:
cos(0) 1
sin(π) 0
tan 2π 0
cos(13π)
(each 2 · π contributes a full revolution.)
Evaluating trigonometric functions.
2 / 13
Last time
Using the techniques we developed last time, evaluate:
cos(0) 1
sin(π) 0
tan 2π 0
cos(13π) -1
(each 2 · π contributes a full revolution.)
Evaluating trigonometric functions.
2 / 13
Last time
Using the techniques we developed last time, evaluate:
cos(0) 1
sin(π) 0
tan 2π 0
cos(13π) -1
(each 2 · π contributes a full revolution.)
We can evaluate the trig functions at any multiple of π/2
Evaluating trigonometric functions.
2 / 13
Last time
Using the techniques we developed last time, evaluate:
cos(0) 1
sin(π) 0
tan 2π 0
cos(13π) -1
(each 2 · π contributes a full revolution.)
We can evaluate the trig functions at any multiple of π/2
Can we evaluate the trig functions at any other angles?
Evaluating trigonometric functions.
2 / 13
Last time
Using the techniques we developed last time, evaluate:
cos(0) 1
sin(π) 0
tan 2π 0
cos(13π) -1
(each 2 · π contributes a full revolution.)
We can evaluate the trig functions at any multiple of π/2
Can we evaluate the trig functions at any other angles?
Today’s Goal: compute the trig functions at π/4, π/6, π/3 and “related”
angles.
Evaluating trigonometric functions.
2 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
Evaluating trigonometric functions.
3 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
If you draw a ray emanating from (0, 0) at an angle
of π/4 radians, you are drawing the line y = x.
Evaluating trigonometric functions.
3 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
If you draw a ray emanating from (0, 0) at an angle
of π/4 radians, you are drawing the line y = x.
Thus, if a rotation of angle π/4 produces the point
(x, y ), then x = y .
Evaluating trigonometric functions.
3 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
If you draw a ray emanating from (0, 0) at an angle
of π/4 radians, you are drawing the line y = x.
Thus, if a rotation of angle π/4 produces the point
(x, y ), then x = y .
sin(π/4) = cos(π/4) = x.
Evaluating trigonometric functions.
3 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
If you draw a ray emanating from (0, 0) at an angle
of π/4 radians, you are drawing the line y = x.
Thus, if a rotation of angle π/4 produces the point
(x, y ), then x = y .
sin(π/4) = cos(π/4) = x.
But sin(π/4)2 + cos(π/4)2 = 1
Evaluating trigonometric functions.
3 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
If you draw a ray emanating from (0, 0) at an angle
of π/4 radians, you are drawing the line y = x.
Thus, if a rotation of angle π/4 produces the point
(x, y ), then x = y .
sin(π/4) = cos(π/4) = x.
But sin(π/4)2 + cos(π/4)2 = 1 So 2x 2 = 1
Evaluating trigonometric functions.
3 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
If you draw a ray emanating from (0, 0) at an angle
of π/4 radians, you are drawing the line y = x.
Thus, if a rotation of angle π/4 produces the point
(x, y ), then x = y .
sin(π/4) = cos(π/4) = x.
√
But sin(π/4)2 + cos(π/4)2 = 1 So 2x 2 = 1 and x = ± √12 = ± 22 .
Evaluating trigonometric functions.
3 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
If you draw a ray emanating from (0, 0) at an angle
of π/4 radians, you are drawing the line y = x.
Thus, if a rotation of angle π/4 produces the point
(x, y ), then x = y .
sin(π/4) = cos(π/4) = x.
√
But sin(π/4)2 + cos(π/4)2 = 1 So 2x 2 = 1 and x = ± √12 = ± 22 .
In which quadrant are we?
Evaluating trigonometric functions.
3 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
If you draw a ray emanating from (0, 0) at an angle
of π/4 radians, you are drawing the line y = x.
Thus, if a rotation of angle π/4 produces the point
(x, y ), then x = y .
sin(π/4) = cos(π/4) = x.
√
But sin(π/4)2 + cos(π/4)2 = 1 So 2x 2 = 1 and x = ± √12 = ± 22 .
In which quadrant are we? Quadrant 1. So sine and cosine are positive.
Evaluating trigonometric functions.
3 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
If you draw a ray emanating from (0, 0) at an angle
of π/4 radians, you are drawing the line y = x.
Thus, if a rotation of angle π/4 produces the point
(x, y ), then x = y .
sin(π/4) = cos(π/4) = x.
√
But sin(π/4)2 + cos(π/4)2 = 1 So 2x 2 = 1 and x = ± √12 = ± 22 .
In which quadrant are we? Quadrant
1. So sine and cosine are positive.
√
2
Thus cos(π/4) = sin(π/4) = 2 .
Evaluating trigonometric functions.
3 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
If you draw a ray emanating from (0, 0) at an angle
of π/4 radians, you are drawing the line y = x.
Thus, if a rotation of angle π/4 produces the point
(x, y ), then x = y .
sin(π/4) = cos(π/4) = x.
√
But sin(π/4)2 + cos(π/4)2 = 1 So 2x 2 = 1 and x = ± √12 = ± 22 .
In which quadrant are we? Quadrant
1. So sine and cosine are positive.
√
2
Thus cos(π/4) = sin(π/4) = 2 .
The other trig functions:
tan(π/4)
cot(π/4)
sec(π/4)
csc(π/4)
Evaluating trigonometric functions.
3 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
If you draw a ray emanating from (0, 0) at an angle
of π/4 radians, you are drawing the line y = x.
Thus, if a rotation of angle π/4 produces the point
(x, y ), then x = y .
sin(π/4) = cos(π/4) = x.
√
But sin(π/4)2 + cos(π/4)2 = 1 So 2x 2 = 1 and x = ± √12 = ± 22 .
In which quadrant are we? Quadrant
1. So sine and cosine are positive.
√
2
Thus cos(π/4) = sin(π/4) = 2 .
The other trig functions:
tan(π/4) = y /x =
√
2/2
√
2/2
=1
cot(π/4)
sec(π/4)
csc(π/4)
Evaluating trigonometric functions.
3 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
If you draw a ray emanating from (0, 0) at an angle
of π/4 radians, you are drawing the line y = x.
Thus, if a rotation of angle π/4 produces the point
(x, y ), then x = y .
sin(π/4) = cos(π/4) = x.
√
But sin(π/4)2 + cos(π/4)2 = 1 So 2x 2 = 1 and x = ± √12 = ± 22 .
In which quadrant are we? Quadrant
1. So sine and cosine are positive.
√
2
Thus cos(π/4) = sin(π/4) = 2 .
The other trig functions:
tan(π/4) = y /x =
cot(π/4) = x/y =
√
2/2
√
2/2
√
2/2
√
2/2
=1
=1
sec(π/4)
csc(π/4)
Evaluating trigonometric functions.
3 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
If you draw a ray emanating from (0, 0) at an angle
of π/4 radians, you are drawing the line y = x.
Thus, if a rotation of angle π/4 produces the point
(x, y ), then x = y .
sin(π/4) = cos(π/4) = x.
√
But sin(π/4)2 + cos(π/4)2 = 1 So 2x 2 = 1 and x = ± √12 = ± 22 .
In which quadrant are we? Quadrant
1. So sine and cosine are positive.
√
2
Thus cos(π/4) = sin(π/4) = 2 .
The other trig functions:
tan(π/4) = y /x =
cot(π/4) = x/y =
sec(π/4) = 1/x =
√
2/2
√
2/2
√
2/2
√
2/2
√1
2/2
=1
=1
√
= 2
csc(π/4)
Evaluating trigonometric functions.
3 / 13
Trig functions at π/4
π/4 radians is half of a right angle.
If you draw a ray emanating from (0, 0) at an angle
of π/4 radians, you are drawing the line y = x.
Thus, if a rotation of angle π/4 produces the point
(x, y ), then x = y .
sin(π/4) = cos(π/4) = x.
√
But sin(π/4)2 + cos(π/4)2 = 1 So 2x 2 = 1 and x = ± √12 = ± 22 .
In which quadrant are we? Quadrant
1. So sine and cosine are positive.
√
2
Thus cos(π/4) = sin(π/4) = 2 .
The other trig functions:
tan(π/4) = y /x =
cot(π/4) = x/y =
sec(π/4) = 1/x =
csc(π/4) = 1/y =
√
2/2
√
2/2
√
2/2
√
2/2
√1
2/2
√1
2/2
=1
=1
√
= 2
√
= 2
Evaluating trigonometric functions.
3 / 13
Using this to fill in the unit circle
We have π/4 on the unit circle.
(1,0)
√
√
2π/3 π/3
( 2/2, 2/2)
3π/4
π
π/4
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
5π/4
3π
2
4π/3 5π/3
(1,0)
7π/4
(0,-1)
Evaluating trigonometric functions.
4 / 13
Using this to fill in the unit circle
We have π/4 on the unit circle.
How can we get to 3π/4?
(1,0)
√
√
2π/3 π/3
( 2/2, 2/2)
3π/4
π
π/4
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
5π/4
3π
2
4π/3 5π/3
(1,0)
7π/4
(0,-1)
Evaluating trigonometric functions.
4 / 13
Using this to fill in the unit circle
We have π/4 on the unit circle.
How can we get to 3π/4? Reflect across the y -axis.
(1,0)
√
√
2π/3 π/3
( 2/2, 2/2)
3π/4
π
π/4
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
5π/4
3π
2
4π/3 5π/3
(1,0)
7π/4
(0,-1)
Evaluating trigonometric functions.
4 / 13
Using this to fill in the unit circle
We have π/4 on the unit circle.
How can we get to 3π/4? Reflect across the y -axis.
(1,0)
√
√
(− 2/2, 2/2)
√
√
2π/3 π/3
( 2/2, 2/2)
3π/4
π
π/4
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
5π/4
3π
2
4π/3 5π/3
(1,0)
7π/4
(0,-1)
Evaluating trigonometric functions.
4 / 13
Using this to fill in the unit circle
We have π/4 on the unit circle.
How can we get to 3π/4? Reflect across the y -axis.
What about 5π/4 and 7π/4?
(1,0)
√
√
(− 2/2, 2/2)
√
√
2π/3 π/3
( 2/2, 2/2)
3π/4
π
π/4
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
5π/4
3π
2
4π/3 5π/3
(1,0)
7π/4
(0,-1)
Evaluating trigonometric functions.
4 / 13
Using this to fill in the unit circle
We have π/4 on the unit circle.
How can we get to 3π/4? Reflect across the y -axis.
What about 5π/4 and 7π/4? Reflect over the x-axis.
(1,0)
√
√
(− 2/2, 2/2)
√
√
2π/3 π/3
( 2/2, 2/2)
3π/4
π
π/4
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
5π/4
3π
2
4π/3 5π/3
(1,0)
7π/4
(0,-1)
Evaluating trigonometric functions.
4 / 13
Using this to fill in the unit circle
We have π/4 on the unit circle.
How can we get to 3π/4? Reflect across the y -axis.
What about 5π/4 and 7π/4? Reflect over the x-axis.
(1,0)
√
√
(− 2/2, 2/2)
√
√
2π/3 π/3
( 2/2, 2/2)
3π/4
π
π/4
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
(1,0)
3π
2
5π/4
7π/4 √
√
√
√
4π/3 5π/3
(− 2/2, − 2/2)
( 2/2, − 2/2)
(0,-1)
Evaluating trigonometric functions.
4 / 13
The other trig functions
(1,0)
√
√
(− 2/2, 2/2)
√
√
2π/3 π/3
( 2/2, 2/2)
3π/4
π
π/4
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
(1,0)
3π
2
5π/4
7π/4 √
√
√
√
4π/3 5π/3
(− 2/2, − 2/2)
( 2/2, − 2/2)
(0,-1)
For you: Compute tan(3π/4), sec(5π/4) and cos(−π/4).
Evaluating trigonometric functions.
5 / 13
Trig functions at π/6.
Next we aim for the trig functions at π/6.
We’ll begin with some geometry.
Evaluating trigonometric functions.
6 / 13
Trig functions at π/6.
Next we aim for the trig functions at π/6.
We’ll begin with some geometry.
Draw a unit circle and a triangle with
vertices (0, 0), (cos(π/6), sin(π/6))
and (cos(π/6), 0).
Evaluating trigonometric functions.
6 / 13
Trig functions at π/6.
Next we aim for the trig functions at π/6.
We’ll begin with some geometry.
Draw a unit circle and a triangle with
vertices (0, 0), (cos(π/6), sin(π/6))
and (cos(π/6), 0).
What are the angles in this triangle?
Evaluating trigonometric functions.
6 / 13
Trig functions at π/6.
Next we aim for the trig functions at π/6.
We’ll begin with some geometry.
Draw a unit circle and a triangle with
vertices (0, 0), (cos(π/6), sin(π/6))
and (cos(π/6), 0).
What are the angles in this triangle?
One is π/6. The other is a right angle.
Evaluating trigonometric functions.
6 / 13
Trig functions at π/6.
Next we aim for the trig functions at π/6.
We’ll begin with some geometry.
Draw a unit circle and a triangle with
vertices (0, 0), (cos(π/6), sin(π/6))
and (cos(π/6), 0).
What are the angles in this triangle?
One is π/6. The other is a right angle.
The angles of a triangle sum to π.
What is the other angle?
Evaluating trigonometric functions.
π/6
π/2
6 / 13
Trig functions at π/6.
Next we aim for the trig functions at π/6.
We’ll begin with some geometry.
Draw a unit circle and a triangle with
vertices (0, 0), (cos(π/6), sin(π/6))
and (cos(π/6), 0).
What are the angles in this triangle?
One is π/6. The other is a right angle.
The angles of a triangle sum to π.
What is the other angle?
Evaluating trigonometric functions.
π/3
π/6
π/2
6 / 13
Trig functions at π/6.
Next we aim for the trig functions at π/6.
We’ll begin with some geometry.
Draw a unit circle and a triangle with
vertices (0, 0), (cos(π/6), sin(π/6))
and (cos(π/6), 0).
What are the angles in this triangle?
One is π/6. The other is a right angle.
The angles of a triangle sum to π.
What is the other angle?
Reflect the triangle over the x-axis and
get another triangle.
Evaluating trigonometric functions.
π/3
π/6
π/6
π/2
π/2
π/3
6 / 13
Trig functions at π/6.
Next we aim for the trig functions at π/6.
We’ll begin with some geometry.
Draw a unit circle and a triangle with
vertices (0, 0), (cos(π/6), sin(π/6))
and (cos(π/6), 0).
What are the angles in this triangle?
One is π/6. The other is a right angle.
The angles of a triangle sum to π.
What is the other angle?
Reflect the triangle over the x-axis and
get another triangle.
What do you get when you put these
triangles together?
Evaluating trigonometric functions.
π/3
π/6
π/6
π/2
π/2
π/3
6 / 13
Trig functions at π/6.
Glue these triangles together.
π/3
π/6
π/6
Evaluating trigonometric functions.
π/2
π/2
π/3
7 / 13
Trig functions at π/6.
Glue these triangles together.
You get an equilateral triangle. All of
the sides are the same length!
π/3
π/3
π/3
Evaluating trigonometric functions.
7 / 13
Trig functions at π/6.
Glue these triangles together.
You get an equilateral triangle. All of
the sides are the same length!
Two of the faces run from the center
of the unit circle to its edge. They are
length one
π/3
π/3
π/3
Evaluating trigonometric functions.
7 / 13
Trig functions at π/6.
Glue these triangles together.
You get an equilateral triangle. All of
the sides are the same length!
Two of the faces run from the center
of the unit circle to its edge. They are
length one
The
rightmost
edge
runs
from
(cos(π/6), sin(π/6))
to
(cos(π/6), − sin(π/6)). Its length is
2 sin(π/6)
Evaluating trigonometric functions.
π/3
π/3
π/3
7 / 13
Trig functions at π/6.
Glue these triangles together.
You get an equilateral triangle. All of
the sides are the same length!
Two of the faces run from the center
of the unit circle to its edge. They are
length one
The
rightmost
edge
runs
from
(cos(π/6), sin(π/6))
to
(cos(π/6), − sin(π/6)). Its length is
2 sin(π/6)
2 sin(π/6) = 1.
Evaluating trigonometric functions.
π/3
π/3
π/3
7 / 13
Trig functions at π/6.
Glue these triangles together.
You get an equilateral triangle. All of
the sides are the same length!
Two of the faces run from the center
of the unit circle to its edge. They are
length one
The
rightmost
edge
runs
from
(cos(π/6), sin(π/6))
to
(cos(π/6), − sin(π/6)). Its length is
2 sin(π/6)
2 sin(π/6) = 1.
sin(π/6) = 21 .
Evaluating trigonometric functions.
π/3
π/3
π/3
7 / 13
Trig functions at π/6.
Glue these triangles together.
You get an equilateral triangle. All of
the sides are the same length!
Two of the faces run from the center
of the unit circle to its edge. They are
length one
The
rightmost
edge
runs
from
(cos(π/6), sin(π/6))
to
(cos(π/6), − sin(π/6)). Its length is
2 sin(π/6)
2 sin(π/6) = 1.
sin(π/6) = 21 .
π/3
π/3
π/3
sin(π/6) = 1/2. cos(π/6) > 0. What is cos(π/6)? The other trig
functions?
Evaluating trigonometric functions.
7 / 13
Angles related to π/6
So far this is what we have.
(1,0)
√
√
(− 2/2, 2/2)
√
√
2π/3 π/3
( 2/2, 2/2)
√
3π/4
π
π/4
( 3/2, 1/2)
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
(1,0)
3π
2
5π/4
7π/4 √
√
√
√
4π/3 5π/3
(− 2/2, − 2/2)
( 2/2, − 2/2)
(0,-1)
Evaluating trigonometric functions.
8 / 13
Angles related to π/6
So far this is what we have.
How can we get an angle of π/3?
(1,0)
√
√
(− 2/2, 2/2)
√
√
2π/3 π/3
( 2/2, 2/2)
√
3π/4
π
π/4
( 3/2, 1/2)
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
(1,0)
3π
2
5π/4
7π/4 √
√
√
√
4π/3 5π/3
(− 2/2, − 2/2)
( 2/2, − 2/2)
(0,-1)
Evaluating trigonometric functions.
8 / 13
Angles related to π/6
So far this is what we have.
How can we get an angle of π/3? Reflect an angle of π/6 over the line
y = x.
(1,0)
√
√
(− 2/2, 2/2)
√
√
2π/3 π/3
( 2/2, 2/2)
√
3π/4
π
π/4
( 3/2, 1/2)
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
(1,0)
3π
2
5π/4
7π/4 √
√
√
√
4π/3 5π/3
(− 2/2, − 2/2)
( 2/2, − 2/2)
(0,-1)
Evaluating trigonometric functions.
8 / 13
Angles related to π/6
So far this is what we have.
How can we get an angle of π/3? Reflect an angle of π/6 over the line
y = x.
(1,0)
√
(1/2,√ 3/2)
√
2π/3 π/3
( 2/2, 2/2)
√
3π/4
π
π/4
( 3/2, 1/2)
2
5π/6
π/6
√
√
(− 2/2, 2/2)
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
(1,0)
3π
2
5π/4
7π/4 √
√
√
√
4π/3 5π/3
(− 2/2, − 2/2)
( 2/2, − 2/2)
(0,-1)
Evaluating trigonometric functions.
8 / 13
Angles related to π/6
So far this is what we have.
How can we get an angle of π/3? Reflect an angle of π/6 over the line
y = x.
2π/3 and 5π/6?
(1,0)
√
(1/2,√ 3/2)
√
2π/3 π/3
( 2/2, 2/2)
√
3π/4
π
π/4
( 3/2, 1/2)
2
5π/6
π/6
√
√
(− 2/2, 2/2)
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
(1,0)
3π
2
5π/4
7π/4 √
√
√
√
4π/3 5π/3
(− 2/2, − 2/2)
( 2/2, − 2/2)
(0,-1)
Evaluating trigonometric functions.
8 / 13
Angles related to π/6
So far this is what we have.
How can we get an angle of π/3? Reflect an angle of π/6 over the line
y = x.
2π/3 and 5π/6?
(1,0)
√
√
(1/2,√ 3/2)
√ (−1/2,
√ 3/2)
√
2π/3 π/3
(− 2/2, 2/2)
( 2/2, 2/2)
√
√
3π/4
π
π/4
(− 3/2, 1/2)
( 3/2, 1/2)
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
(1,0)
3π
2
5π/4
7π/4 √
√
√
√
4π/3 5π/3
(− 2/2, − 2/2)
( 2/2, − 2/2)
(0,-1)
Evaluating trigonometric functions.
8 / 13
Angles related to π/6
So far this is what we have.
How can we get an angle of π/3? Reflect an angle of π/6 over the line
y = x.
2π/3 and 5π/6? 7π/6, 4π/3, 5π/3, 11π/6?
(1,0)
√
√
(1/2,√ 3/2)
√ (−1/2,
√ 3/2)
√
2π/3 π/3
(− 2/2, 2/2)
( 2/2, 2/2)
√
√
3π/4
π
π/4
(− 3/2, 1/2)
( 3/2, 1/2)
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
(1,0)
3π
2
5π/4
7π/4 √
√
√
√
4π/3 5π/3
(− 2/2, − 2/2)
( 2/2, − 2/2)
(0,-1)
Evaluating trigonometric functions.
8 / 13
Angles related to π/6
So far this is what we have.
How can we get an angle of π/3? Reflect an angle of π/6 over the line
y = x.
2π/3 and 5π/6? 7π/6, 4π/3, 5π/3, 11π/6?
(1,0)
√
√
(1/2,√ 3/2)
√ (−1/2,
√ 3/2)
√
2π/3 π/3
(− 2/2, 2/2)
( 2/2, 2/2)
√
√
3π/4
π
π/4
(− 3/2, 1/2)
( 3/2, 1/2)
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
7π/6
(1,0)
3π
2
5π/4
7π/4 √
√
√
√
4π/3 5π/3
(− 2/2, − 2/2)
( 2/2, − 2/2)
(0,-1)
Evaluating trigonometric functions.
8 / 13
Angles related to π/6
So far this is what we have.
How can we get an angle of π/3? Reflect an angle of π/6 over the line
y = x.
2π/3 and 5π/6? 7π/6, 4π/3, 5π/3, 11π/6?
(1,0)
√
√
(1/2,√ 3/2)
√ (−1/2,
√ 3/2)
√
2π/3 π/3
(− 2/2, 2/2)
( 2/2, 2/2)
√
√
3π/4
π
π/4
(− 3/2, 1/2)
( 3/2, 1/2)
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
(1,0)
7π/6
√
(− 3/2, −1/2) 5π/4 3π
2
7π/4 √
√
√
√
4π/3 5π/3
(− 2/2, − 2/2)
( 2/2, − 2/2)
(0,-1)
Evaluating trigonometric functions.
8 / 13
Angles related to π/6
So far this is what we have.
How can we get an angle of π/3? Reflect an angle of π/6 over the line
y = x.
2π/3 and 5π/6? 7π/6, 4π/3, 5π/3, 11π/6?
(1,0)
√
√
(1/2,√ 3/2)
√ (−1/2,
√ 3/2)
√
2π/3 π/3
(− 2/2, 2/2)
( 2/2, 2/2)
√
√
3π/4
π
π/4
(− 3/2, 1/2)
( 3/2, 1/2)
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
(1,0)
7π/6
√
(− 3/2, −1/2) 5π/4 3π
2
7π/4 √
√
√
√
4π/3 5π/3
(− 2/2, − 2/2)
( 2/2, − 2/2)
√
(−1/2, − 3/2)
(0,-1)
Evaluating trigonometric functions.
8 / 13
Angles related to π/6
So far this is what we have.
How can we get an angle of π/3? Reflect an angle of π/6 over the line
y = x.
2π/3 and 5π/6? 7π/6, 4π/3, 5π/3, 11π/6?
(1,0)
√
√
(1/2,√ 3/2)
√ (−1/2,
√ 3/2)
√
2π/3 π/3
(− 2/2, 2/2)
( 2/2, 2/2)
√
√
3π/4
π
π/4
(− 3/2, 1/2)
( 3/2, 1/2)
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
(1,0)
7π/6
√
(− 3/2, −1/2) 5π/4 3π
2
7π/4 √
√
√
√
4π/3 5π/3
(− 2/2, − 2/2)
( 2/2,
√
√ − 2/2)
(−1/2, − 3/2)
(1/2, − 3/2)
(0,-1)
Evaluating trigonometric functions.
8 / 13
Angles related to π/6
So far this is what we have.
How can we get an angle of π/3? Reflect an angle of π/6 over the line
y = x.
2π/3 and 5π/6? 7π/6, 4π/3, 5π/3, 11π/6?
(1,0)
√
√
(1/2,√ 3/2)
√ (−1/2,
√ 3/2)
√
2π/3 π/3
(− 2/2, 2/2)
( 2/2, 2/2)
√
√
3π/4
π
π/4
(− 3/2, 1/2)
( 3/2, 1/2)
2
5π/6
π/6
(-1,0)
π radians
0 radians
2π radians
11π/6
(1,0)
7π/6
√
√
(− 3/2, −1/2) 5π/4 3π
(
−1/2)
2
7π/4 √ 3/2,√
√
√
4π/3
(− 2/2, − 2/2)
( 2/2,
5π/3
√
√ − 2/2)
(−1/2, − 3/2)
(1/2, − 3/2)
(0,-1)
Evaluating trigonometric functions.
8 / 13
The identities we used: How are θ, 2π − θ, π − θ and
π + θ related?
Here are an angle of θ and an angle
of 2π − θ.
Does any simple geometry relate
them?
θ
2π − θ
−θ
Evaluating trigonometric functions.
9 / 13
The identities we used: How are θ, 2π − θ, π − θ and
π + θ related?
Here are an angle of θ and an angle
of 2π − θ.
Does any simple geometry relate
them?
Reflection over the x-axis
θ
2π − θ
−θ
Evaluating trigonometric functions.
9 / 13
The identities we used: How are θ, 2π − θ, π − θ and
π + θ related?
(x, y )
Here are an angle of θ and an angle
of 2π − θ.
Does any simple geometry relate
them?
Reflection over the x-axis
θ
2π − θ
−θ
(x, −y )
Evaluating trigonometric functions.
9 / 13
The identities we used: How are θ, 2π − θ, π − θ and
π + θ related?
Here are an angle of θ and an angle
of 2π − θ.
Does any simple geometry relate
them?
Reflection over the x-axis
If cos(θ) = x and sin(θ) = y
then cos(2π − θ) = x and sin(2π −
θ) = −y
This proves a useful identity:
(x, y )
θ
2π − θ
−θ
Evaluating trigonometric functions.
(x, −y )
9 / 13
The identities we used: How are θ, 2π − θ, π − θ and
π + θ related?
Here are an angle of θ and an angle
of 2π − θ.
Does any simple geometry relate
them?
Reflection over the x-axis
If cos(θ) = x and sin(θ) = y
then cos(2π − θ) = x and sin(2π −
θ) = −y
This proves a useful identity:
(x, y )
θ
2π − θ
−θ
(x, −y )
Theorem
For any angle θ, cos(−θ) = cos(2π − θ) = cos(θ) and
sin(−θ) = sin(2π − θ) = − sin(θ)
What are the corresponding identities about tan, sec and csc?
Evaluating trigonometric functions.
9 / 13
θ and π + θ?
θ
Here are an angle of θ and an angle of π + θ.
Does any simple geometry relate them?
π
θ
Evaluating trigonometric functions.
10 / 13
θ and π + θ?
θ
Here are an angle of θ and an angle of π + θ.
Does any simple geometry relate them?
Reflection across the origin
π
θ
Evaluating trigonometric functions.
10 / 13
θ and π + θ?
(x, y )
θ
Here are an angle of θ and an angle of π + θ.
Does any simple geometry relate them?
Reflection across the origin
π
θ
(−x, −y )
Evaluating trigonometric functions.
10 / 13
θ and π + θ?
(x, y )
Here are an angle of θ and an angle of π + θ.
Does any simple geometry relate them?
Reflection across the origin
If cos(θ) = x and sin(θ) = y
then cos(π + θ) = −x and sin(pi + θ) = −y
θ
π
θ
(−x, −y )
Evaluating trigonometric functions.
10 / 13
θ and π + θ?
(x, y )
Here are an angle of θ and an angle of π + θ.
Does any simple geometry relate them?
Reflection across the origin
If cos(θ) = x and sin(θ) = y
then cos(π + θ) = −x and sin(pi + θ) = −y
θ
π
θ
(−x, −y )
This proves a useful identity:
Theorem
For any angle θ, cos(pi + θ) = − cos(θ) and sin(π + θ) = − sin(θ)
Evaluating trigonometric functions.
10 / 13
θ and π + θ?
(x, y )
Here are an angle of θ and an angle of π + θ.
Does any simple geometry relate them?
Reflection across the origin
If cos(θ) = x and sin(θ) = y
then cos(π + θ) = −x and sin(pi + θ) = −y
θ
π
θ
(−x, −y )
This proves a useful identity:
Theorem
For any angle θ, cos(pi + θ) = − cos(θ) and sin(π + θ) = − sin(θ)
For you: What are the corresponding identities about tan, sec and csc?
Evaluating trigonometric functions.
10 / 13
θ and π − θ?
We could derive the formula relating the trig functions of θ and π − θ in
the same way.
Evaluating trigonometric functions.
11 / 13
θ and π − θ?
We could derive the formula relating the trig functions of θ and π − θ in
the same way. Instead we’ll use algebra to derive it.
Compute:
cos(π − θ) = cos(π + (−θ))
Evaluating trigonometric functions.
11 / 13
θ and π − θ?
We could derive the formula relating the trig functions of θ and π − θ in
the same way. Instead we’ll use algebra to derive it.
Compute:
cos(π − θ) = cos(π + (−θ)) What does adding π do?
=
Evaluating trigonometric functions.
11 / 13
θ and π − θ?
We could derive the formula relating the trig functions of θ and π − θ in
the same way. Instead we’ll use algebra to derive it.
Compute:
cos(π − θ) = cos(π + (−θ)) What does adding π do?
= − cos(−θ)
Evaluating trigonometric functions.
11 / 13
θ and π − θ?
We could derive the formula relating the trig functions of θ and π − θ in
the same way. Instead we’ll use algebra to derive it.
Compute:
cos(π − θ) = cos(π + (−θ)) What does adding π do?
= − cos(−θ)
What is cos(−θ)?
=
Evaluating trigonometric functions.
11 / 13
θ and π − θ?
We could derive the formula relating the trig functions of θ and π − θ in
the same way. Instead we’ll use algebra to derive it.
Compute:
cos(π − θ) = cos(π + (−θ)) What does adding π do?
= − cos(−θ)
What is cos(−θ)?
= − cos(θ)
Evaluating trigonometric functions.
11 / 13
θ and π − θ?
We could derive the formula relating the trig functions of θ and π − θ in
the same way. Instead we’ll use algebra to derive it.
Compute:
cos(π − θ) = cos(π + (−θ)) What does adding π do?
= − cos(−θ)
What is cos(−θ)?
= − cos(θ)
For you: Compute:
sin(π − θ) = sin(π + (−θ)) =
Evaluating trigonometric functions.
11 / 13
θ and π − θ?
We could derive the formula relating the trig functions of θ and π − θ in
the same way. Instead we’ll use algebra to derive it.
Compute:
cos(π − θ) = cos(π + (−θ)) What does adding π do?
= − cos(−θ)
What is cos(−θ)?
= − cos(θ)
For you: Compute:
sin(π − θ) = sin(π + (−θ)) = sin(θ)
Evaluating trigonometric functions.
11 / 13
θ and π − θ?
We could derive the formula relating the trig functions of θ and π − θ in
the same way. Instead we’ll use algebra to derive it.
Compute:
cos(π − θ) = cos(π + (−θ)) What does adding π do?
= − cos(−θ)
What is cos(−θ)?
= − cos(θ)
For you: Compute:
sin(π − θ) = sin(π + (−θ)) = sin(θ)
Theorem
cos(π − θ) = − cos(θ) while sin(π − θ) = sin(θ)
Evaluating trigonometric functions.
11 / 13
θ and
π
2
− θ?
θ
Here are an angle of θ and an angle of π2 − θ.
Does any simple geometry relate them?
π
2
Evaluating trigonometric functions.
−θ
12 / 13
θ and
π
2
− θ?
θ
Here are an angle of θ and an angle of π2 − θ.
Does any simple geometry relate them?
π/2
Evaluating trigonometric functions.
π
2
−θ
12 / 13
θ and
π
2
− θ?
(x, y )
θ
Here are an angle of θ and an angle of π2 − θ.
Does any simple geometry relate them?
π/2
Evaluating trigonometric functions.
π
2
−θ
12 / 13
θ and
π
2
− θ?
(x, y )
Here are an angle of θ and an angle of π2 − θ.
Does any simple geometry relate them?
θ
π/2 θ π2 − θ
Evaluating trigonometric functions.
12 / 13
θ and
π
2
− θ?
(x, y )
Here are an angle of θ and an angle of π2 − θ.
Does any simple geometry relate them?
Reflection across the line y = x
θ
π/2 θ π2 − θ
Evaluating trigonometric functions.
12 / 13
θ and
π
2
− θ?
(x, y )
Here are an angle of θ and an angle of π2 − θ.
Does any simple geometry relate them?
Reflection across the line y = x
θ
(y , x)
π/2 θ π2 − θ
Evaluating trigonometric functions.
12 / 13
θ and
π
2
− θ?
(x, y )
Here are an angle of θ and an angle of π2 − θ.
Does any simple geometry relate them?
Reflection across the line y = x
If cos(θ) = x and sin(θ) = y
then cos(π/2 − θ) = y and sin(π/2 − θ) = x
Evaluating trigonometric functions.
θ
(y , x)
π/2 θ π2 − θ
12 / 13
θ and
π
2
− θ?
(x, y )
Here are an angle of θ and an angle of π2 − θ.
Does any simple geometry relate them?
Reflection across the line y = x
If cos(θ) = x and sin(θ) = y
then cos(π/2 − θ) = y and sin(π/2 − θ) = x
θ
(y , x)
π/2 θ π2 − θ
This proves a useful identity:
Theorem
For any angle θ, cos(π/2 − θ) = sin(θ) and sin(π/2 − θ) = cos(θ)
Evaluating trigonometric functions.
12 / 13
θ and
π
2
− θ?
(x, y )
Here are an angle of θ and an angle of π2 − θ.
Does any simple geometry relate them?
Reflection across the line y = x
If cos(θ) = x and sin(θ) = y
then cos(π/2 − θ) = y and sin(π/2 − θ) = x
θ
(y , x)
π/2 θ π2 − θ
This proves a useful identity:
Theorem
For any angle θ, cos(π/2 − θ) = sin(θ) and sin(π/2 − θ) = cos(θ)
For you: What are the corresponding identities about tan, sec and csc?
Evaluating trigonometric functions.
12 / 13
With any remaining time we should go through the remaining spaces in
the handout.
Evaluating trigonometric functions.
13 / 13