Navigation Review
• A passenger in an airplane flying at
an altitude of 10km sees 2 towns
directly to the east of the plane.
The angles of depression to the
towns are 28o and 55o. How far
apart are the towns?
• 11.81 miles
One More
• A plane is 160 miles north and 85
miles east of an airport. The pilot
wants to fly directly to the airport.
What bearing should be taken?
• N 27.98o E
Basic Trigonometric
Identities
2.4
JMerrill, 2006
Revised, 2009
(contributions from DDillon)
Trig Identities
Identity: an equation that is true for
all values of the variable for which
the expressions are defined
Recall
y
sin  
r
x
cos  
r
y
tan  
x
r
csc 
y
r
sec 
x
x
cot  
y
Recall
•
3
If sec x  , and tan x  0, find the other
2
trig functions
 5
sin x 
3
2
cos x 
3
3 3 5
csc x 

5
5
3
sec x 
2
5
tan x 
2
2
2 5
cot x 

5
5
-2
 5
3
Fundamental
Trigonometric Identities
Reciprocal Identities
sin  
1
csc
1
cos 
sec
1
tan  
cot 
Also true:
1
csc 
sin 
1
sec 
cos
1
cot  
tan 
Using the Reciprocal
Identities
• If sin θ = x, find csc θ.
1
csc  
x
• If
2 3
csc  
3
find sin θ
3
sin  
2
Fundamental
Trigonometric Identities
Quotient Identities
sin 
tan  
cos
cos
cot  
sin 
Using the Quotient
Identities
4
3
• If sin   , cos   , find tan  and cot 
5
5
4
sin  5 4  5  4
tan  
   
cos  3 5  3  3
5
3
cos  5 3  5  3
cot  
   
sin  4 5  4  4
5
Is there an easier way to find cotθ?
Notation
• We will now begin looking at squared
trig functions.
• (sinθ)2 is written sin2θ. We write the
square over the word because we are
not squaring the angle (sinθ2), we are
squaring the sine of the angle.
Fundamental
Trigonometric Identities
Pythagorean Identities
sin   cos   1
2
2
1  cot   csc 
2
2
tan   1  sec 
2
2
These are crucial!
You MUST memorize
them.
You will not be able to
do any of our future
work if you do not
know them!
Pythagorean Memory
Trick
sin2
cos2
1
tan2
sec2
cot2
csc2
(Add the top of the triangle to = the bottom)
Different Forms
Pythagorean Identities appear in
other forms
sin 2   cos 2   1
sin   1  cos 
1  cot   csc 
cot   csc   1
tan   1  sec 
tan   sec   1
2
2
2
2
2
2
2
2
2
2
Example
4
• Find cosθ and tanθ if sinθ = and θ
5
is in QII.
• First we will use Identities:
Example
sin 2   cos 2   1
2
4
sin 
4
5
tan  


cos  3 3
5
4
2
   cos   1
5
16
 cos 2   1
25
9
2
cos  
25
9
cos   
25
3
3
cos    , cos  is negative in QII , so cos  
5
5
Example
• Same problem—using triangles:
4
• Find cosθ and tanθ if sinθ = and θ
5
is in QII.
x 3
cos   
r
5
y 4
tan   
x
3
4
5
-3
Simplifying #1
cot x sin x
cos x

sin x
sin x
cos x

sin x
sin x
 cos x
Simplifying #2
1  cos x
2
cos x
2
2
sin x

2
cos x
 tan x
2
Simplifying - You Do
• Simplify tanx cscx
sec x
One More Thing
• The test is next Tuesday.
• Work through the Chapter Reviews and Practice
Tests
• Know the Identities
• Know your trig functions in terms of x,y,r
• Know the exact value from the chart on P39 (sin,
cos, tan)
• Email me with any questions over the weekend.
I’ll check email!
• I’ll put power points and lab on my website on
Saturday.
• The lab is due next Thursday—not Tuesday.