Name:_______________________
Date assigned:______________
Band:________
Precalculus | Packer Collegiate Institute
Cofunctions and an Introduction to Trigonometric Identities
Section 0: Warm Up
What are the three Pythagorean trigonometric identities?
Section 1: Cofunctions
On the left is a right triangle. Draw in the right angle symbol at the right
angle.
One angle is labeled  (the Greek letter alpha).
In the diagram, write in the value of the other angle.
Now we’re going to come up with some relationships. You are going to be given the basic trig functions of
How do they relate to the trig functions of the other angle?
(a) sin   _________________
&
cos   _________________
(b) tan   _________________
&
cot   _________________
(c) sec   _________________
&
csc   _________________
 below.
Because of these beautiful relationships, these trig functions are called “co-functions.”
(Ever notice COsine, COtangent, COsecant?)
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Introductory Problems:
So if I wanted to rewrite sin(20o ) as a cosine, I could simply write: cos(_____)
And if I wanted to rewrite csc( /11) as a secant, I could simply write: sec(_____)
Practice Problems (from Section 8.1 #19-28): Find the exact value of the expression without a calculator
1.
cos10o
sin 80o
3. 1  tan 2 5o  csc 2 85o
2. 1  cos 2 20o  cos 2 70o
4. sec 35o csc 55o  tan 35o cot 55o
5. What transformations does the parent function y  cos( x) undergo when graphing y  cos(90o  x) ?
6. Below is a graph of y  cos( x) . Put it through the transformations you listed above to graph y  cos(90o  x) .
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Section 2: Introduction to Trigonometric Identities
Everyone at your table should take out your graphing calculators. Did one of you forget it? TEN LASHES WITH A WET
NOODLE.
Each person should take a different one of these equations and enter it into Y1. DON’T GRAPH IT THOUGH. Double check
your equation to make sure you didn’t mis-enter anything or forget a parentheses!
Now BEING SURE TO HIDE YOUR GRAPH FROM THE REST OF YOUR MISCREANT GROUP MEMBERS use ZOOM TRIG and
graph. THIS IS FOR YOUR EYES ONLY.
A.
C.


y  tan  x    1
4

2cos( x)
y
cos( x)  sin( x)


y  2 cos( x) sec  x  
4

1  tan( x)
D. y  1 
1  tan( x)
B.
Okay, are you all graphed? On the count of three there will be the big reveal.
One…
TWO…
REVEAL!
Whoa. Really? REALLY?
Yes, really.
The reason this is happening is because: _________________________________________________________________
__________________________________________________________________________________________________.
1. Think independently (not in your group). What do you suspect will happen if you graph: y 
sin 2 ( x)
1  cos 2 ( x)
2. Everyone in your group share your conjecture.
3. Everyone in the group should graph this function (ZOOM TRIG!). You can enter sin 2 ( x) by either typing
(sin( x))(sin( x)) or (sin( x))^2 . Are you surprised by what you see? Now that you see it, can you explain what
is showing up?
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What this all is showing you is that sometimes two functions are equal even if their equations look very different.
They are identical to each other. You’ve seen this a million times before…
For example: y  ( x 2  3) 2 and y  x 4  6 x 2  9 are the exact same when graphed. The points which make one
true also are the same points that make the second one true. The way you can see that is by simple algebraic
manipulation.
We call two expressions which look different but are essentially the same “identities.” Duh, they are identical. Since
we are dealing with trigonometry, we call identities with trigonometry (this is really shocking… ready?):
trigonometric identities.
Check yo’self: Graph the following on geogebra and see if you can’t generate simple trigonometric identities:
(a) y  tan( x) cot( x) is equivalent to _________________________________
(b) y 
cot x
is equivalent to _________________________________
csc x
(c) y  sec( x) sin( x) is equivalent to _________________________________
Check yo’self twice:
Are the following two equations identical? Graph them on geogebra to find out.
(a) y 
1  sin( x)
cos( x)

cos( x) 1  sin( x)
(b) y  ln tan x
(c) y  4(3) 2
(d) y 
(csc x  1)(sec x  tan x)
csc x sec x
sin 3 x  cos3 x
sin x  cos x
and
y  tan x ?
YES / NO
and
y  ln sin x  ln cos x ?1
YES / NO
and
y  (2  3cos x) 2 ?
YES / NO
and
y  sin x cos x ?
YES / NO
As we go through the next few days, we will be working to verify some identities are true.
Home Enjoyment:
Section 8.1#19, 20, 22, 25, 26, 27
Section 7.3 #9-18 (all) [Example 1 from the section should help you]
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Absolute value in geogebra is as follows: abs(x)
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