Transcript Trigonometric Identities - Phoenix Union High School District

5
Trigonometric
Identities
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5 Trigonometric Identities
5.1 Fundamental Identities
5.2 Verifying Trigonometric Identities
5.3 Sum and Difference Identities for Cosine
5.4 Sum and Difference Identities for Sine
and Tangent
5.5 Double-Angle Identities
5.6 Half-Angle Identities
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5.1 Fundamental Identities
Fundamental Identities ▪ Using the Fundamental Identities
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Fundamental Identities
Reciprocal Identities
Quotient Identities
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Fundamental Identities
Pythagorean Identities
Negative-Angle Identities
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Note
In trigonometric identities, θ can be
an angle in degrees, a real number,
or a variable.
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Example 1
If
FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
QUADRANT
and θ is in quadrant II, find each function
value.
Pythagorean
identity
(a) sec θ
In quadrant II, sec θ is negative, so
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Example 1
FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
QUADRANT (continued)
(b) sin θ
Quotient identity
Reciprocal identity
from part (a)
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Example 1
FINDING TRIGONOMETRIC FUNCTION
VALUES GIVEN ONE VALUE AND THE
QUADRANT (continued)
(b) cot(– θ)
Reciprocal identity
Negative-angle
identity
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Caution
To avoid a common error, when
taking the square root, be sure to
choose the sign based on the
quadrant of θ and the function being
evaluated.
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Example 2
EXPRESSING ONE FUNCITON IN
TERMS OF ANOTHER
Write cos x in terms of tan x.
Since sec x is related to both cos x and tan x by
identities, start with
Take reciprocals.
Reciprocal identity
Take the square
root of each side.
 1  tan2 x
 cos x
2
1  tan x
The sign depends on
the quadrant of x.
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Example 3
REWRITING AN EXPRESSION IN
TERMS OF SINE AND COSINE
1  cot 2 
Write
in
terms
of
sin
θ
and
cos
θ,
and
1  csc 2 
then simplify the expression so that no quotients
appear.
cos2 
1  cot 2  1  sin2 

2
1  csc  1  1
sin2 
 cos2  
2
1

sin

2


sin  

1 

2
1

sin


2 
sin  
Quotient identities
Multiply numerator
and denominator by
the LCD.
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REWRITING AN EXPRESSION IN
TERMS OF SINE AND COSINE (cont’d)
Example 3
sin2   cos2 

sin2   1
Distributive property
1

2
 cos 
Pythagorean identities
  sec 2 
Reciprocal identity
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Caution
When working with trigonometric
expressions and identities, be sure
to write the argument of the function.
For example, we would not write
An argument such as θ
is necessary in this identity.
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