Download Section 5.1 Fundamental Identities

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TRIG-Fall 2011-Jordan
Trigonometry, 9th edition, Lial/Hornsby/Schneider, Pearson, 2009
Chapter 5: Trigonometric Identities
Section 5.1
Fundamental Identities
Negative-Angle Identities
sin (-θ) = - sin θ
csc (-θ) = -csc θ
cos (-θ) = cos θ
sec (-θ) = sec θ
tan (-θ) = -tan θ
cot (-θ) = -cot θ
Reciprocal Identities
sin  
1
csc 
csc  
1
sin 
cos 
1
sec 
sec  
1
cos
tan  
1
cot 
cot  
1
tan 
Pythagorean Identities
Quotient Identities
sin2   cos2   1
tan  
sin 
cos
tan2   1  sec2 
cot  
cos
sin 
1  cot 2   csc2 
Example 1
values.
a) sec θ
b) sin θ
c) cot (-θ)
If tan θ = -5/3 and θ is in quadrant II, find each of the following function
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Example 2
Express sec x in terms of sin x
Example 3 First rewrite cot 2 θ(1 + tan 2 θ) in terms of sin θ and cos θ and then
simplify so that no quotients appear in the final expression.
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Section 5.2
Verifying Trigonometric Identities
Verifying Trigonometric Identities
An identity is an equation that is true for all its domain values.
To verify an identity, we show that one side of the identity can be rewritten in a form that
is identical to the other side.
Guidelines for Verifying Trigonometric Identities
Know the fundamental identities and their equivalent forms, e.g., sin2 x + cos2 x = 1 is
equivalent to cos2 x = 1 – sin2 x.
Start working with the more complicated side of the equation and try to turn it into the
simpler side.
Perform any indicated operations such as adding fractions, squaring binomials,
distributing, or factoring.
Sometimes it is helpful to express all trigonometric functions on one side of an equation
in terms of sine and cosine.
Sometimes it is helpful to rewrite one side of the equation in terms of a single
trigonometric function.
Multiplying both the numerator and denominator of a fraction by the same factor (usually
the conjugate of the numerator or denominator) may help.
As you selection substitutions, keep in mind the side you are not changing. It
represents your goal. Look for the identity or function which best links the two sides.
If you get really stuck, abandon the side you’re working on and start working on the
other side. Try to make the two sides “meet in the middle.”
Caution: Verifying identities is not the same as solving equations. Techniques used in
solving equations, such as adding the same term to both sides or multiplying both sides
by the same factor, are not valid when verifying identities.
Example 1
Perform the indicated operation and simplify the result:
cos
sin 

sin  1  cos
Example 2
Factor the expression:
cot 4 x  3 cot 2 x  2
csc  sec 
cot 
Example 3
Simplify:
Example 4
Verify the identity:
1
 csc x  sin x
sec x tan x
Example 5
Verify the identity:
1
1

 2 csc2 x
1  cos x 1  cos x
Example 6
Verify the identity:
sin x
 csc x  cot x
1 cos x
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Section 5.3
Sum and Difference Identities for Cosine
Cosine of a Sum or Difference
cos( A  B)  cos A cos B  sin A sin B
cos( A  B)  cos A cos B  sin A sin B
Caution:
cos (A + B)  cos A + cos B, etc.
Example 1
5
a) cos
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Find the exact value.
b) cos 95 cos 35 + sin 95 sin 35
Example 2 Given sin s = 24/25 and cos t = -4/5, s in Quadrant II and t in Quadrant III,
find
a) cos (s + t)
b) cos (s – t)
Cofunction Identities
sin (90 - ) = cos θ
csc (90 - ) = sec θ
cos (90 - ) = sin θ
sec (90 - ) = csc θ
tan (90 - ) = cot θ
cot (90 - ) = tan θ
If θ is in radian measure, then replace 90° with π/2.
Example 3
Prove the cofunction identity cos (90 - ) = sin θ
Example 4
Write sin
5
in terms of the cofunction of a complementary angle.
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Section 5.4
Sum and Difference Identities for Sine and Tangent
Sine of a Sum or Difference
sin( A  B)  sin A cos B  cos Asin B
sin( A  B)  sin A cos B  cos Asin B
Caution:
sin (A + B)  sin A + sin B, etc.
Tangent of a Sum or Difference
tan( A  B) 
Example 1
a) sin
b)

tan A  tan B
1  tan A tan B
tan( A  B) 
tan A  tan B
1  tan A tan B
Find the exact value.
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tan(105)
c) sin 90° cos 135° - cos 90° sin 135°
Example 2


Use an identity to write sin    as a single function of θ.
6

Example 3
sin s = 4/5 and cos t = -5/13, s in quadrant II and t in quadrant III
a) Find sin (s + t)
b) Find tan (s + t)
c) Find the quadrant of s + t
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Section 5.5
Double-Angle Identities
Double-Angle Identities
Caution:
sin 2A ≠ 2 sin A
Example 1 Given
a) find sin θ
or
cos 2 A
 cos A , etc.
A
and θ terminates in quadrant III,
b) find cos θ
Example 2 Given
a) find sin 2θ
and sin θ < 0,
b) find cos 2θ
c) find tan 2θ
Example 3 Use an identity to write
value or as a single number.
Example 4
Verify the following identity:
as a single trigonometric function
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Product-to-Sum Identities
Caution:
sin M sin N ≠ sin (MN), etc.
Example 5
functions.
Write the following expression as a sum or difference of trigonometric
2 cos 85° sin 140°
Sum-to-Product Identities
Caution:
sin M – sin N ≠ sin (M – N), etc.
(You cannot factor out or distribute a trig function)
Example 6
Write the following expression as the product of two functions.
cos 2θ – cos 4θ
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Section 5.6
Half-Angle Identities
Half-Angle Identities
Caution:
The plus or minus signs of sin (A/2) and cos (A/2) are selected according
to the quadrant in which A/2 terminates.
Example 1
Find the exact value of cos 112.5°
Example 2
Use an identity to write the following expression as a single trigonometric
function.
Example 3 Given that sin θ = -7/25 with 180° < θ < 270°,
a) find the exact value of sin (θ/2)
b) find the exact value of cos (θ/2)
c) find the exact value of tan (θ/2)