Download Notes Section 6.1 Reciprocal, Quotient, and Pythagorean Identities

Pre-Calculus 12A
Section 6.1
Reciprocal, Quotient, and Pythagorean Identities
In mathematics, it is important to understand the difference between an equation and an identity.

An equation is only true for certain value(s) of the variable. For example, the equation x2  4  0 is true
only for x  2 .

An identity is an equation that is true for all values of the variable. For example, the equation
( x  1)2  x2  2 x  1 is true for all values of x, so we can call it an identity.
A trigonometric identity is a trigonometric equation that is true for all permissible values of the variable.

An example of a trigonometric equation is sin x  1 , which is true only for x  90o  360o n, n W .

An example of a trigonometric identity is sin 2 x  cos2 x  1 , which is true for all values of x.

An example of a trigonometric identity with non-permissible values is csc x 
1
, which is true for all
sin x
values of x except x  0o  180o n, n W , since sin x  0 for these values of x.
RECIPROCAL AND QUOTIENT IDENTITES
We are already familiar with a few trigonometric identities, referred to as the reciprocal and quotient
identities.
RECIPROCAL IDENTITIES
sin  
1
csc 
cos  
1
sec 
tan 
1
cot 
csc  
1
sin 
sec  
1
cos 
cot  
1
tan
QUOTIENT IDENTITIES
tan 
sin
cos 
cot  
cos 
sin 
We will be using these and other fundamental trigonometric identities to prove other trigonometric identities.
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Pre-Calculus 12A
Section 6.1
Example 1: Verify a Potential Identity Numerically
Consider the equation sec  
tan 
.
sin 
a. Verify that the equation sec  
tan 
is valid for   60 .
sin 
LHS = sec  sec60o 
RHS =
tan 
tan 60o


sin 
sin 60o
Therefore, the equation is true for   60 .
Note: Verifying that an equation is valid for a particular value of the variable is not sufficient to
conclude that the equation is an identity.
b. Prove that sec  
tan 
is an identity.
sin 
Note: To prove that an identity is true for all permissible values, express both sides of the identity in
equivalent forms. One or both sides of the identity must be algebraically manipulated into an equivalent
form to match the other side.
There is a major difference between solving a trigonometric equation and proving a trigonometric
identity:

Solving a trigonometric equation determines the value(s) that make that particular equation true.
You perform operations across the = sign to solve for the variable.

Proving a trigonometric identity shows that the expressions on each side of the = sign are
equivalent. You work on each side of the identity independently, and you do not perform
operations across the = sign.
LHS
sec
RHS
tan 
sin 
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Pre-Calculus 12A
Section 6.1
c. Determine the non-permissible values, in degrees, for the identity sec  
tan 
.
sin 
Note: Non-permissible values of the variable are values for which denominators are zero or numerators
are undefined.
To determine the non-permissible values, assess each trigonometric function in the equation individually.
LHS:
sec is undefined when when  = ____________________________.
RHS:
tan  is undefined when  = ____________________________.
sin   0 when  = ____________________________.
Combined, the non-permissible values for are:  = _____________________________________________.
Example 2: Use Identities to Simplify Expressions
Simplify the expression
tan x cos x
.
sec x cot x
To simplify the expression, we can use the reciprocal and quotient identities to rewrite the expression in terms
of sine and cosine.
tan x cos x
=
sec x cot x
Example 3: Using the Reciprocal and Quotient Identities
Prove each of the following identities for all permissible values of x.
tan x cos x
1
sin x
LHS
tan x cos x
sin x
cot x  sec x 
RHS
1
LHS
cot x  sec x
cos2 x  sin x
sin x cos x
RHS
cos x  sin x
sin x cos x
2
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Pre-Calculus 12A
Section 6.1
The Pythagorean Identity
Development of the Pythagorean Identity:
In the diagram to the left note the Pythagorean Theorem:
a2  b 2  c 2 .
Replace these values with x, y and r.
____________________________________
In the unit circle r = 1, cos   x and sin  y , substitute these
values for x, y and r.
THE PYTHAGOREAN
IDENTITY__________________________________________
cos 2   sin2   1
cos 2  
sin2  
cos2   sin2   1
cos2   sin2   1
Divide each term by cos 
Divide each term by sin2 
Now isolate each term
individually to find the other
versions of the identity.
Now isolate each term
individually to find the other
versions of the identity.
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THE PYTHAGOREAN IDENTITIES
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Pre-Calculus 12A
Section 6.1
Example 4: Using the Pythagorean Identities
Prove each of the following identities for all permissible values of x.
a. cos x  sin2 x cos x  cos3 x
b. sec x  cos x  tan x sin x
c. tan x sin x  sec x  cos x
sec x
d. sin x 
cot x  tan x
LHS
sin x
b. sec x  cos x  tan x sin x
RHS
cos x  tan x sin x
Solution:
a. cos x  sin2 x cos x  cos3 x
LHS
RHS
2
cos x  sin x cos x
cos3 x
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Pre-Calculus 12A
Section 6.1
c. tan x sin x  sec x  cos x
LHS
tan x sin x
RHS
sec x  cos x
d. sin x 
sec x
cot x  tan x
sin x
sec x
cot x  tan x
Strategies That May Assist in Proving Trigonometric Identities
1. Start by working on whatever side looks more complicated. Then work toward the less complicated
expression.
2. Rewrite everything in terms of sine and cosine.
3. If you have fractions, think about making common denominators.
4. Factor expressions to cancel out terms or create other identities.
5. Expand expressions to create identities.
6. When there is a square in the term, look for a way to apply the Pythagorean Identity.
7. Always keep the “target expression” in mind. Continually refer to it.
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Pre-Calculus 12A
Section 6.1
Example 5: Solve Equations Using the Quotient, Reciprocal, and Pythagorean Identities
Solve for  in general form and then in the specified domain. Use exact solutions when possible. Check for
non-permissible values.
a. cos2   cot  sin , 0    2
b. 2sin  7  3csc  ,  360    180
c.
sin2   2cos   2,  2    2
Solution:
a.
cos2   cot  sin , 0    2
b. 2sin  7  3csc  ,  360    180
c.
sin2   2cos   2,  2    2
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