Transcript Algebra and Trigonometry - North Carolina Central University

Verifying Trigonometric Identities
Dr .Hayk Melikyan
Departmen of Mathematics and CS
[email protected]
H.Melikyan/1200
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Basic Trigonometric Identities
Reciprocal Identities
1
csc x = sin x
1
sec x = cos x
1
cot x = tan x
Quotient Identities
sin x
tan x = cos x
cos x
cot x = sin x
Pythagorean Identities
sin2x + cos2x = 1
tan2x + 1 = sec2x
1 + cot2x = csc2x
Even-Odd Identities
sin(- x ) = - sin x
sec( -x ) = sec x
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cos(- x ) = cosx
csc( -x ) = - cscx
tan( -x) = - tanx
cot( -x) = - cotx
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Text Example
Verify the identity: sec x cot x = csc x.
Solution
The left side of the equation contains the more complicated
expression. Thus, we work with the left side. Let us express this side of the
identity in terms of sines and cosines. Perhaps this strategy will enable us to
transform the left side into csc x, the expression on the right.
1
cosx
sec x cotx =

cosx sin x
1
=
= csc x
sin x
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Apply a reciprocal identity: sec x = 1/cos x
and a quotient identity: cot x = cos x/sin x.
Divide both the numerator and the
denominator by cos x, the common factor.
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Text Example
Verify the identity: cosx - cosxsin2x = cos3x
Solution
We start with the more complicated side, the left side. Factor out
the greatest common factor, cos x, from each of the two terms.
cos x - cos x sin2 x = cos x(1 - sin2 x)
= cos x ·
= cos3 x
cos2
x
Factor cos x from the two terms.
Use a variation of sin2 x + cos2 x = 1.
Solving for cos2 x, we obtain cos2 x =
1 – sin2 x.
Multiply.
We worked with the left and arrived at the right side. Thus, the identity is verified.
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Guidelines for Verifying Trigonometric Identities
1.
Work with each side of the equation independently of the other
side. Start with the more complicated side and transform it in a
step-by-step fashion until it looks exactly like the other side.
2.
Analyze the identity and look for opportunities to apply the
fundamental identities. Rewriting the more complicated side of
the equation in terms of sines and cosines is often helpful.
3.
If sums or differences of fractions appear on one side, use the
least common denominator and combine the fractions.
4.
Don't be afraid to stop and start over again if you are not getting
anywhere.
Creative puzzle solvers know that strategies
leading to dead ends often provide good problem-solving ideas.
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Example
Verify the identity:
csc(x) / cot (x) = sec (x)
Solution:
csc x
= sec x
cot x
1
sin x = 1
cos x
cos x
sin x
1
sin x
1

=
sin x cos x cos x
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Example
Verify the identity:
cos x = cos x  cos x sin x
3
2
Solution:
cos x = cos3 x  cos x sin 2 x
cos x = cos x(cos x  sin x)
2
2
cos x = cos x(1)
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Example
Verify the following identity:
tan2 x - cot2 x
= tan x - cos x
tan x  cot x
Solution:
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sin 2 x cos2 x
2
2
2
tan x - cot x cos x sin 2 x
=
sin x cos x
tan x  cot x

cos x sin x
sin 4 x - cos4 x
4
4
2
2
sin
x
cos
x cos x sin x
cos
x
sin
x
=
=

2
2
2
2
sin x  cos x
cos x sin x
1
cos x sin x
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Example cont.
Solution:
sin 4 x - cos4 x
cos x sin x
(sin 2 x  cos2 x)(sin2 x - cos2 x)
=
sin x cos x
sin 2 x - cos2 x
sin 2 x
cos2 x
=
=
sin x cos x
sin x cos x sin x cos x
sin x cos x
=
= t an x - cot x
cos x sin x
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Sum and Difference Formulas
Dr .Hayk Melikyan
Departmen of Mathematics and CS
[email protected]
H.Melikyan/1200
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The Cosine of the Difference of Two Angles
cos( -  ) = cos cos   sin  sin 
The cosine of the difference of two angles equals the
cosine of the first angle times the cosine of the
second angle plus the sine of the first angle times the
sine of the second angle.
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Text Example

Find the exact value of cos 15°
Solution
We know exact values for trigonometric functions of 60° and 45°.
Thus, we write 15° as 60° - 45° and use the difference formula for cosines.
cos l5° = cos(60° - 45°)
= cos 60° cos 45°  sin 60° sin 45°
1
2
3
2



2 2
2
2
=
=
2
6

4
4
2 6
4
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cos( -) = cos  cos   sin  sin 
Substitute exact values from
memory or use special triangles.
Multiply.
Add.
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Text Example
Find the exact value of ( cos 80° cos 20°  sin 80° sin 20°) .
Solution
The given expression is the right side of the formula for cos( - )
with  = 80° and  = 20°.
cos( -) = cos  cos   sin  sin 
cos 80° cos 20°  sin 80° sin 20° = cos (80° - 20°) = cos 60° = 1/2
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Example

Find the exact value of cos(180º-30º)
Solution
cos(180- 30)
= cos180cos30  sin 180sin 30
3
1
= -1*
 0*
2
2
3
=2
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Example

Verify the following identity:
5 
2

cos x -  = (cosx  sin x)
4 
2

Solution
5 

cos x 
4


 5 
 5 
= cos x cos   sin x sin  
 4 
 4 
2
2
cos x  sin x
2
2
2
=(cos x  sin x )
2
=-
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Sum and Difference Formulas for Cosines and Sines
cos(   ) = cos cos  - sin  sin 
cos( -  ) = cos cos   sin  sin 
sin(   ) = sin  cos   cos sin 
sin( -  ) = sin  cos  - cos sin 
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Example

Find the exact value of sin(30º+45º)
Solution
sin(   ) = sin  cos   cos sin 
sin(30  45) = sin 30 cos 45  cos30sin 45
1
2
3
2
= 


2 2
2
2
2 6
=
4
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Sum and Difference Formulas for Tangents
The tangent of the sum of two angles equals the tangent of the first angle
plus the tangent of the second angle divided by 1 minus their product.
t an  t an 
t an(   ) =
1 - t an t an 
t an - t an 
t an( -  ) =
1  t an t an 
The tangent of the difference of two angles equals the tangent of the first angle
minus the tangent of the second angle divided by 1 plus their product.
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Example

Find the exact value of tan(105º)
Solution
•tan(105º)=tan(60º+45º)
tan  tan 
tan(   ) =
1 - tan tan 
tan60  tan 45
=
1 - tan60 tan 45
3 1 1 3
=
=
1- 3 1- 3
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Example

Write the following expression as the sine, cosine, or tangent of
an angle. Then find the exact value of the expression.
7

7 
sin cos - cos sin
12 12
12 12
Solution
7

7

sin
cos - cos sin
12
12
12
12
6
 7  
= sin 
-  = sin
12
 12 12 
= sin
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
2
=1
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