Download Trigonometric Identities

Module 6 Lesson 4
Using Trigonometric Identities – Part 2
There are more identities for Trigonometry. We won’t be getting into specifics on how
they originated. They are divided into two groups for you. Identities you need to know
how to use and be familiar with and identities that you just need to be aware of their
existence.
Identities you’ll need to know of and how to use:
Sum and Difference IdentitiesSum and difference identities are used to find trigonometric values of the combination
of two angles. Let’s look at the identities:
Sum:
Difference:
sin(   )  sin  cos   cos  sin 
cos(   )  cos  cos   sin  sin 
tan   tan 
tan(   ) 
1  tan  tan 
sin(   )  sin  cos   cos  sin 
cos(   )  cos  cos   sin  sin 
tan   tan 
tan(   ) 
1  tan  tan 
Alpha, α, and Beta, β, represent two different angles. Using these identities is as simple
as plugging values in and doing some basic Algebra.
Example 1:
Given:
sin 20  0.342
cos 20  0.940
and
sin15  0.259
cos15  0.966
Here is the sum identity for sine:
, find the value for the sine of 35◦
sin(   )  sin  cos   cos  sin 
This identity states that the sine value of the sum of these angles is equal to the sine of
the first times the cosine of the second, plus the cosine of the first times the sine of the
second. We simply plug in the values:
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sin(35 )  sin(15  20 )
sin(   )  sin  cos   cos  sin 
sin(15  20 )  sin15 cos 20  cos15 sin 20
sin(15  20 )  (0.259)(0.940)  (0.966)(0.342)
sin(15  20 )  0.24346  0.330372
sin(15  20 )  0.574
sin(35 )  0.574
The important thing to remember is to make sure you get the angles in the correct
order. For the sum identities, recall the addition is commutative, so it doesn’t matter
which angle is Alpha and which is Beta. For difference identities, it does make a
difference:
Example 2:
Given:
sin 20  0.342
cos 20  0.940
and
sin15  0.259
cos15  0.966
, find the value for the sine of 5◦
Since 20 – 15 = 5, we can use the difference identity for sine. 20◦ must be Alpha and 15◦
must be Beta:
sin(5 )  sin(20  15 )
sin(   )  sin  cos   cos  sin 
sin(20  15 )  sin 20 cos15  cos 20 sin15
sin(20  15 )  (0.342)(0.966)  (0.940)(0.259)
sin(20  15 )  0.330372  0.24346
sin(20  15 )  0.086912
sin(5 )  0.0869
Using two of the sum identities, we found the values for sine of 35◦ and 5◦. You can
verify those with a calculator.
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Example 3:
Given:
sin 20  0.342
cos 20  0.940
and
sin15  0.259
cos15  0.966
, find the value for the cosine of 35◦ and 5◦
cos(35 )  cos(15  20 )
cos(   )  cos  cos   sin  sin 
cos(15  20 )  cos15 cos 20  sin15 sin 20
cos(15  20 )  (0.966)(0.940)  (0.259)(0.342)
cos(15  20 )  0.90804  0.088578
cos(15  20 )  0.819462
cos(35 )  0.819
cos(5 )  cos(20  15 )
cos(   )  cos  cos   sin  sin 
cos(20  15 )  cos 20 cos15  sin 20 sin15
cos(20  15 )  (0.940)(0.966)  (0.342)(0.259)
cos(20  15 )  0.90804  0.088578
cos(20  15 )  0.996618
cos(5 )  0.9966
You won’t be required to memorize these identities, but you will be required to know
how to use them.
Double AngleThe other set of identities that you’ll be required to know how to use (and not
memorize!) are the double angle identities.
The double angle identities allow you to find the trigonometric values for an angle twice
as large as an angle that you already know. They are also used to help rewrite and solve
trigonometric equations:
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sin 2  2sin  cos 
cos 2  cos2   sin 2 
cos 2  2 cos2   1
cos 2  1  2sin 2 
2 tan 
tan 2 
1  tan 2 
Example 4:
Say you know:
sin 20  0.342
cos 20  0.940
We can use the double angle formulas to find the sine and cosine of 40 degrees.
Remember for cosine, there are three different formulas and you select the one you
want based on whether or not you know the sine, cosine or both (of the original angle).
In this case, we know both, so we have our choice:
cos 2  2 cos 2   1
sin 2  2sin  cos 
sin(40 )  2(sin 20 )(cos 20 )
sin(40 )  2(0.342)(0.940)
sin(40 )  0.6430
cos(40 )  2(cos 20 ) 2  1
cos(40 )  2(0.940) 2  1
cos(40 )  1.7672  1
cos(40 )  0.7672
Verify those results with your calculator.
Example 6:
Given:
sin16  0.276
cos16  0.961
, find sin 32◦
sin 2  2sin  cos 
sin(32 )  2sin(16 ) cos(16 )
sin(32 )  2(0.276)(0.961)
sin(32 )  0.530
Example 7:
Given: sin 37  0.602 , find cos 74◦
We must use this double angle cosine formula:
since we only know the sine value of 37 degrees.
cos 2  1  2sin 2 
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cos 2  1  2sin 2 
cos(74 )  1  2sin 2 (37 )
cos(74 )  1  2(0.602) 2
cos(74 )  0.275
Example 8:
Given: tan101 = -5.145 , find tan 202◦
2 tan 
tan 2 
1  tan 2 
2 tan(101 )
tan(202 ) 
1  tan 2 (101 )
2( 5.145)
tan(202 ) 
1  ( 5.145) 2
10.29
tan(202 ) 
1  26.471025
10.29
tan(202 ) 
25.471025
tan(202 )  0.404
Identities you’ll just need to know of:
For this section, just make sure you read through the descriptions. You won’t be
required to memorize the actual identities or know how to use them. I just want you to
be aware of them for future reference.
SymmetryThe symmetry identities tell us how the sign of a trigonometric value of sine or cosine
varies with the corresponding angles in the different quadrants. In other words, if the
sine value is negative for an angle, will it be positive or negative 360 degrees later?
What about 180 degrees? What if it’s reflected over the x-axis? What if it’s reflected
over the y-axis?
Here are the identities. We have them for sine and cosine since the four other
trigonometric values can be derived from sine and cosine: (k is any integer)
Case I:
sin( A  360 k )  sin A
cos( A  360 k )  cos A
Case II:
sin( A  180 (2k  1))   sin A
cos( A  180 (2k  1))   cos A
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Case III:
sin(360 k  A)   sin A
Case IV:
cos(360 k  A)  cos A
sin(180 (2k  1)  A)  sin A
cos(180 (2k  1)  A)   cos A
Case I represents the coterminal angle in the same quadrant – the signs of both sine and
cosine stay the same for the new angle.
Case II represents adding 180 degrees to the original angle. When than happens, the
sine and cosine of the new angle both take on the opposite signs they had originally.
Case III represents reflecting the angle across the x-axis. When this occurs, the sign of
the sine value changes, the cosine sign does not change.
Case IV represents reflecting the angle across the y-axis. When this occurs, the sign of
the cosine value changes, but the sine sign does not change.
Remember that the numerical values are all the same, these identities just tell us if the
sign changes or not.
Half AngleLike the double angle identities, the half angle identities are what they sound like. They
allow you to find the trigonometric values for an angle half the value of one already
known. These identities are also used like the double angle, to help rewrite and solve
trigonometric equations:

1  cos 
cos  
2
2
sin
tan
tan
tan

2

2

2

2

1  cos 
2
1  cos 
1  cos 
sin 

1  cos 
1  cos 

sin 

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Sum ProductThis group of identities will be used when you need to rewrite a sum or difference of
two trigonometric values as a product:
cos   cos   2 cos(

) cos(
 
)
2
2

 
cos   cos   2sin(
)sin(
)
2
2

 
sin   sin   2sin(
) cos(
)
2
2

 
sin   sin   2 cos(
)sin(
)
2
2
Product SumThis group of identities will be used when you need to rewrite a trigonometric product
of values as a sum or difference:
cos(   )  cos(   )
2
cos(   )  cos(   )
sin  sin  
2
sin(   )  sin(   )
cos  sin  
2
sin(   )  sin(   )
sin  cos  
2
cos  cos  
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