Download Pythagorean and Quotient Identities Briana J

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Using Pythagorean and Quotient Identities in Trig
Equations
By Briana J.
Period 8
Algebra 2/ Trig
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Quick Review
Before we can get straight into what these identities are and how to use them
we need to do a quick review on the following:
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Right Triangles and labeling accordingly
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Trig Functions
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Reciprocal Trig functions
+ What is a right triangle and how do
I label it correctly?
A right triangle is simply a triangle that has an angle equaling 90°.
Now how do you label it?
a
Hypotenuse
Adjacent
Opposite
There are names for each side of a
right triangle. The longest side is
called the hypotenuse. The side
next to the angle your trying to
find, lets name that angle a, is
labeled as adjacent. Lastly, the
side opposite angle a is labeled
opposite.
+ Trig functions
There are three main trig functions you should know, those are Sine, Cosine, and
Tangent
Sine:
Opposite
θ
Adjacent
Cosine:
Adjacent
Hypotenuse
Adjacent
Hypotenuse
Tangent: Opposite
Adjacent
Opposite
A good way to remember these trig functions is using the acronym SOHCAHTOA
+ Reciprocal Trig Functions
Just like there are trig functions there are reciprocal trig functions. The three you will need
to know are Cosecant, Secant, and Cotangent. They can all be expressed two ways.
Cosecant:
Hypotenuse
Opposite
Secant:
OR
Hypotenuse
Adjacent
Cotangent:
1
θ
1
OR
Adjacent
Opposite
Sin
Cosine
1
OR
H
A
O
Tangent
I like to write the acronym CSCSECCOT under SOHCAHTOA to remember which function
is the opposite of the other.
+ Pythagorean Identities
What is a Pythagorean Identities?
There are three Pythagorean Identities we will be using.
*Note: There are variations that pop up so make sure you know them.
They are:
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Sin2 θ+ Cos2θ=1 OR Sin2 θ=1- Cos2 θ OR Cos2 θ= 1- Sin2 θ
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1+ Tan2 θ= Sec2 OR Tan2 θ= Sec2 θ-1
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1+ Cot2 θ=Csc2 θ OR Cot2 θ= Csc2 θ-1
+ Using Pythagorean Identities
Pythagorean Identities are used to find missing Trigonometric values and simplifying
trigonometric expressions.
Take a look at this example:
If Csc θ= 5/3 and tan θ= 3/4, Find the remaining trig functions.
First lets draw a triangle and label it
Lets identify our Trig Functions. Tangent is Opposite
over Adjacent. Cosecant is Hypotenuse over opposite,
which is the opposite of sine. So the hypotenuse is 5,
the Adjacent is 4, and the Opposite is 3.
A=4
Now lets label θ
Now that we have our Triangle labeled we can solve for the
missing trig function which is Cosine.
Cos θ= 4/5
Cos2 θ= 1- Sin2 θ
Cos2 θ= 1- 16/25
Cos2 θ= 1- (3/5)2 θ
Cos2 θ= 16/25
H=5
θ
O=3
Since all values are positive this
triangle must be in Quadrant 1.
+ Quotient Identities
There are two quotient Identities you need to know.
Tan θ= Sin θ
Cos θ
AND
Cot θ= Cos θ
Sin θ
They are used to find tangent and the opposite of Tangent which is
Cotangent
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Using Quotient Identities
Here is a practice problem:
If Cos θ= 5/13 and Sin θ= 12/13, What is the value of Tan θ?
First lets write the formula we will be using
Tan θ= Sin θ
Cos θ
Now lets plug in our values
Tan θ= Sin θ
Cos θ
12
13
5
13
Multiply the Numerator and the
Denominator by the reciprocal
12
13
13 X 5
5 X 13
13
5
Tan θ= 12/5
*After simplifying
+ Using Identities in Trig Equations
Now lets practice what we learned. This is a trig equation. We have to solve it using the
identities we learned.
2 Cos2x + 3 Sin x- 3 = 0
This equation uses a Pythagorean Identity.
Now Factor and Solve
It uses the Identity Sin2 x + Cos2 x = 1. Now lets plug it in.
2 Cos2x + 3 Sin x- 3 = 0
2(1- Sin2x)+ 3 sin x- 3=0
Now simplify
2(1- Sin2x)+ 3 sin x- 3=0
2-2 sin2x+ 3 sin x-3=0
-2 sin2x + 3sin x-1=0
Multiply by -1
-1(-2 sin2x + 3sin x-1)=0
2 sin2 x – 3 sin x + 1=0
2 sin2 x – 3 sin x + 1=0
(2 sin x -1)(sin x – 1)=0
2 sin x= 0 OR sin x – 1=0
2 sin x= 1
sin x = 1
Sin x = ½
X= π/6, 5π/6 or x= π/2
+ Practice Links
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http://www.regentsprep.org/Regents/math/algtrig/ATT10/
trigequations2.htm
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http://www.regentsprep.org/Regents/math/algtrig/ATT9/pythpractice.htm
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http://www.pc.maricopa.edu/ctlt/titleV/Math/RelatingTrigFunctions/
RelatingTrigFunctions_1_63.html