RIGHT

1
TRIGOMOMETRY
RIGHT
R I A N G L E
2
In a right triangle, the shorter sides are called
legs and the longest side (which is the one
opposite the right angle) is called the hypotenuse
Well label them a, b, and c and the angles ? and
?. Trigonometric functions are defined by taking
the ratios of sides of a right triangle.
?
hypotenuse
c

First lets look at the three basic functions.
b
leg
SINE
COSINE
?
TANGENT
leg
a
They are abbreviated using their first 3 letters
3
We could ask for the trig functions of the angle
? by using the definitions.
You MUST get them memorized. Here is a mnemonic
to help you.
?
c

The old Indian word
b
SOHCAHTOA
SOHCAHTOA
?
INE
a
ANGENT
OSINE
PPOSITE
DJACENT
DJACENT
PPOSITE
YPOTENUSE
YPOTENUSE
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It is important to note WHICH angle you are
talking about when you find the value of the trig
function.
?
Let's try finding some trig functions with some
numbers. Remember that sides of a right triangle
follow the Pythagorean Theorem so
hypotenuse
c
5
b
4
opposite
?
a
3
Let's choose
sin ?
Use a mnemonic and figure out which sides of the
triangle you need for sine.
Use a mnemonic and figure out which sides of the
triangle you need for tangent.
tan ?
5
You need to pay attention to which angle you want
the trig function of so you know which side is
opposite that angle and which side is adjacent to
it. The hypotenuse will always be the longest
side and will always be opposite the right angle.
This method only applies if you have a right
triangle and is only for the acute angles (angles
less than 90) in the triangle.
?
5
4
?
3
6
There are three more trig functions. They are
called the reciprocal functions because they are
reciprocals of the first three functions.
Oh yeah, this means to flip the fraction over.
Like the first three trig functions, these are
referred to by the first three letters except for
cosecant since it's first three letters are the
same as for cosine.
Best way to remember these is learn which is
reciprocal of which and flip them.
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Which trig function is this the reciprocal of?
?
so
c
5
b
4
cot ?
?
a
3
so
As a way to help keep them straight I think, The
"s" doesn't go with "s" and the "c" doesn't go
with "c" so if we want secant, it won't be the
one that starts with an "s" so it must be the
reciprocal of cosine. (have to just remember
that tangent cotangent go together but this
will help you with sine and cosine).
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We can discover the quotient identities if we
take quotients of sin and cos
Remember to simplify complex fractions you invert
and multiply (take the bottom fraction and "flip"
it over and multiply to the top fraction).
Which trig function is this?
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Now to discover my favorite trig identity, let's
start with a right triangle and the Pythagorean
Theorem.
Rewrite trading terms places
Divide all terms by c2
c2
c2
c2
Move the exponents to the outside
This one is sin
This one is cos
Look at the triangle and the angle ? and
determine which trig function these are.
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This is a short-hand way you can write trig
functions that are squared
Now to find the two more identities from this
famous and oft used one.
Divide all terms by cos2?
cos2?
cos2?
cos2?
What trig function is this squared?
1
What trig function is this squared?
Divide all terms by sin2?
sin2?
sin2?
sin2?
These three are sometimes called the Pythagorean
Identities since they come from the Pythagorean
Theorem
1
What trig function is this squared?
What trig function is this squared?
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RECIPROCAL IDENTITIES
QUOTIENT IDENTITIES
PYTHAGOREAN IDENTITIES
All of the identities we learned are found in the
back page of your book under the heading
Trigonometric Identities and then Fundamental
Identities.
You'll need to have these memorized or be able to
derive them for this course.
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If the angle ? is acute (less than 90) and you
have the value of one of the six trigonometry
functions, you can find the other five.
Reciprocal of sine so "flip" sine over
Sine is the ratio of which sides of a right
triangle?
When you know 2 sides of a right triangle you can
always find the 3rd with the Pythagorean theorem.
Now find the other trig functions
?
3
a
"flipped" cos
1
"flipped" tan
Draw a right triangle and label ? and the sides
you know.
13
There is another method for finding the other 5
trig functions of an acute angle when you know
one function. This method is to use fundamental
identities.
We'd still get csc by taking reciprocal of sin
Now use my favorite trig identity
Sub in the value of sine that you know
Solve this for cos ?
This matches the answer we got with the other
method
square root both sides
We won't worry about ? because angle not negative
You can easily find sec by taking reciprocal of
cos.
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Let's list what we have so far
We need to get tangent using fundamental
identities.
Simplify by inverting and multiplying
Finally you can find cot by taking the reciprocal
of this answer.
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SUMMARY OF METHODS FOR FINDING THE REMAINING 5
TRIG FUNCTIONS OF AN ACUTE ANGLE, GIVEN ONE TRIG
FUNCTION.
METHOD 1
1. Draw a right triangle labeling ? and the two
sides you know from the given trig function.
2. Find the length of the side you don't know by
using the Pythagorean Theorem.
3. Use the definitions (remembered with a
mnemonic) to find other basic trig functions.
4. Find reciprocal functions by "flipping" basic
trig functions.
METHOD 2
Use fundamental trig identities to relate what
you know with what you want to find subbing in
values you know.
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The sum of all of the angles in a triangle always
is 180
90
What is the sum of ? ??
Since we have a 90 angle, the sum of the other
two angles must also be 90 (since the sum of all
three is 180).
Two angles whose sum is 90 are called
complementary angles.
?
c
opposite ?
adjacent to ?
b
?
a
Since ? and ? are complementary angles and sin ?
cos ?, sine and cosine are called cofunctions.
adjacent to ?
opposite ?
This is where we get the name cosine, a
cofunction of sine.
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Looking at the names of the other trig functions
can you guess which ones are cofunctions of each
other?
secant and cosecant
tangent and cotangent
Let's see if this is right. Does sec ? csc ??
hypotenuse over adjacent
hypotenuse over opposite
?
c
opposite ?
adjacent to ?
b
?
a
This whole idea of the relationship between
cofunctions can be stated as
adjacent to ?
opposite ?
Cofunctions of complementary angles are equal.
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Cofunctions of complementary angles are equal.
cos 27
sin(90 - 27)
sin 63
Using the theorem above, what trig function of
what angle does this equal?
Let's try one in radians. What trig functions of
what angle does this equal?
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We can't use fundamental identities if the trig
functions are of different angles.
Use the cofunction theorem to change the
denominator to its cofunction
Now that the angles are the same we can use a
trig identity to simplify.