Transcript 5-4C: Special Angles

Calculus Prerequisite Knowledge
0.2: Trignometric Functions
Objective:
•Understand relationships
between angles and sides of
right triangles
•Review trig functions of
special & quadrantal angles
©2008 Roy L. Gover (www.mrgover.com)
Definition
The special angles are:
•30°,60°, and 45° in degrees
•  ,  , and  in radians
6
3
4
In calculus, we use radians.
Important Idea
These angles are special
because they have exact
value trig functions.
Consider the first two
special angles…

6
Long Side
Short Side

3
…orientation
does not
change the
relationships
between
sides and
angles
Short Side

3

6
…orientation
does not
change the
relationships
between
sides and
angles

6

3
Short Side
Analysis
Consider
an
equilateral
triangle
with sides
2 units
long
2
2
2
Analysis
...equilateral
triangles are
equiangular 2
and each

angle must

3
be 3

3
2
2

3
Analysis
…the angle
bisector
2
intersects
the base at
the midpoint 
3
1
 
6 6
2

3
1
Analysis
2

3
1
?
 
6 6
2

3
1

6

3
2
1
?= 3
Try This
Find the
height of an
equilateral
triangle with
each side 4
inches long.
4
4
?
4
Important Idea
In a
  
,
,
6 3 2
rad triangle:
•the short side is one-half
the hypotenuse.
•the long side is
the short side.
3 times
Try This
Find the

length of
6
the
5
3
missing
sides:
10
5
Try This
Find the
length of
the
missing
sides:
5 3
3
5

3
10 3
3
Try This
What is
wrong
with this
picture?

6
Consider the last special
angle…

6

3

4

4

4
…orientation
does not
change the
relationships
between
sides and
angles

4

4
…orientation
does not
change the
relationships
between
sides and
angles

4

4
…and
sides
x
opposite

equal
4
angles are
 x
equal...
x 2
4
and by the
pythagorean theorem, the
hypotenuse is...
Important Idea
  
In a 4 , 4 , 2 triangle:
•The legs of the triangle
are equal.
•the hypotenuse is 2
times the length of the leg.
Try This
Find
the
length
of the
missing
sides
2 2

4
2

4
2
Try This
Find
the
length
of the
missing
sides
2

4
2

4
2
Review
y
opposite
• sin   
r hypotenuse
•
x
adjacent
cos   
r hypotenuse
1
r

-1
y
x
-1
•
y opposite
tan   
x adjacent
Memorize
1
Review
r hypotenuse
• csc  
y
opposite
•
r hypotenuse
sec  
x
adjacent
1
r

-1
y
x
-1
•
x adjacent
cot   
y opposite
Memorize
1
Example
Find the

exact
values of 3
the six trig
functions of the
given special angle
in standard position.
1
-1
1
-1
Analysis


3
rads
1
1. Make a
right
triangle.

-1
-1
1
Analysis


3
rads
1
2. Let the short
2
3
side be 1, then

use your
1
knowledge of
special angles
to find the measures of each
side of the right triangle.
-1
1
-1
Analysis
3. Use the
definitions of
the trig
functions to
find each trig
function exact
value.


3
rads
1
2

1
-1
-1
3
1
Analysis
opp
3
sin  

hyp
2
adj 1
cos  

hyp 2


3
rads
1
2

1
-1
-1
opp
3
tan  

 3
adj
1
3
1
Analysis
hyp 2 3
csc  

opp
3
hyp
sec  
2
adj

3
rads
1
2

1
-1
-1
adj
3
cot  

opp
3

3
1
Try This

Find the
exact
4
values of
the six trig
functions of the
given special angle
in standard position.
1
-1
1
-1
Solution

2
sin 
4
2

2
cos 
4
2
tan

4
1
csc
sec

4

4
cot

4
 2
 2
1
Important Idea
If the given angle is other
than a quadrant I angle,
the exact values of the trig
functions are calculated
using the reference angle
of the given angle.
Review
Reference Angle: the angle
between a given angle and
the nearest x axis. (Note: x
axis; not y axis). Reference
angles are always positive.
Review
How you find the reference
angle depends on which
quadrant contains the
given angle.
Review
2
1
-2
-1
Given
Angle &
Reference
Angle
1
-1
-2
2
3
Review
2
Reference
Angle
-3
-2
1
-1
Given
Angle
1
-1
-2
2
Review
2
Given
Angle
1
-3
-2
-1
1
-1
Reference
Angle
-2
2
Review
2
1
-3
-2
Given
Angle
-1
1
2
-1
-2
Reference
Angle
Example
Draw a
diagram
showing the
relevant angles
and sides and
find the sin,
cos & tan of
5
6
5
6
Given
Angle

Ref. Angle
6
Example
Short Side

6
Long Side
Given
5 Angle
6
Example
2
1

6
 3
5
3
tan

6
3
5
5
3
6
cos

6
2
5 1
sin

6 2
Why is 1 & 2 pos. & 3 neg.?
Try This
Draw a
diagram
showing the
relevant angles
and sides and
find the csc,
sec & cot of
5
6
5
6
Given
Angle

Ref. Angle
6
Solution
2
1

6
 3
5
cot
 3
5
6
6
5
2 3
sec
6

5
csc
2
6
3
Find the Quadrantal Angles
values of


the six trig
functions of 2
the given
csc

sin

angle in
cos sec
standard
position.
tan  cot 
1
-1
1
-1
Find the Quadrantal Angles
values of
the six trig 2
functions of
the given
csc

sin

angle in
cos sec
standard
position.
tan  cot 
1
-1
1
-1
Find the Quadrantal Angles
values of
the six trig
functions of
the given
csc

sin

angle in
cos sec
standard
position.
tan  cot 
1

-1
1
-1
Find the Quadrantal Angles
values of
3
the six trig
2
functions of
the given
csc

sin

angle in
cos sec
standard
position.
tan  cot 
1
-1
1
-1
Important Ideas
•Trig functions of quadrantal
angles have exact values.
•Trig functions of special
angles have exact values.
•Trig functions of all other
angles have approximate
values.
Example
Use a calculator to
 4 
approximate cos   to 4
 9 
decimal places.
You did not forget to
check “mode”, did you???
Example
Use a calculator to
approximate csc(11.55)
to 4 decimal places.
Check “mode” every
time.
Lesson Close
You will frequently use the
trig knowledge you learned
in Precalculus in Calculus
Assignment
Worksheet 0-2