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Trigonometric IdentitiesSum and Differences
WARNING: While viewing this pdf, the viewer may experience the following:
1.) Shock 2.) Confusion 3.) Denial 4.) Disbelief 5.) “I never
learned this”
9.) Indifference 10.) Kanye8.) Terror7.) Rage6.) Fear
Identities FORMULA
sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
sin(α – β) = sin(α)cos(β) – cos(α)sin(β)
cos(α + β) = cos(α)cos(β) – sin(α)sin(β)
cos(α – β) = cos(α)cos(β) + sin(α)sin(β)
tan(α + β) = tan(𝑎) + tan(𝑏)
1 − tan(𝑎)tan(𝑏)
tan(α – β) =tan(a) − tan(b)1 + tan(a)tan(b)
Notice: Only for tan
and cos does the
operation sign change
to it’s opposite
You would need to
frequently refer back to
the Unit Circle while
solving Trigonometric
Identities/ Equations
cos75°
Cos75°
Cos(210-135)
cos210cos135+sin210sin135
−3
2× (−
2
2) + (-
1
2) ×
2
2
ANSWER:
6
4+ (−
2
4)
OR
6 − 2
4
This slide will explain how to solve the equation using the
formula : cos(α – β)
First find two degrees from the Unit Circle that when subtracted equals
the degree give in the problem. For this problem the degree is 75.
We’re going to using the degrees 210 and 135
Next plug in the degrees/numbers in the extended
formula: cos(α)cos(β) + sin(α)sin(β)
α = 210 β= 135
Now, refer back to the Unit Circle
and find the numbers that
correspond with degree. Don’t
forget to change the sign to its
opposite.
Side Note
• Sin is the y
coordinate
• Cos is the
X
coordinate
Multiply from denominator by dominator
and numerator to numerator.
Solve cos75° Cos75°
cos(45+30)
cos45cos30-sin45sin30
(2
2×
3
2) − (
2
2×
1
2)
6
4−
2
4or
6 − 2
4
We’re first going to solve this equation using the formula: cos(α + β)
First, find two numbers that when added together, equals
the number in the problem. For this problem the number is
75. We’re going to use the numbers 45 and 35 degrees.
Next step is plug in the numbers in the
extended formula of cos(α + β) :
Cosαsinβ- SinαCosβ
*Don’t
forget to
change the
operation
from
addition to
subtraction
Next refer back to the Unit Circle
and find number that applies for
the section for example:
Cos45 = 2
2cos30 =
3
2
Sin45 = 2
2Sin30 =
1
2
Multiply the Dominator by Dominator
and Numerator to Numerator for
each section and you got your answer
Sin75°
• Sin75°
• sin(30 + 45)
• 𝑠𝑖𝑛30𝑐𝑜𝑠45 + 𝑐𝑜𝑠30𝑠𝑖𝑛45
•1
2×
2
2+
3
2×
2
2
• ANSWER
•2
4+
6
4
For this problem we’ll being using the formula:
sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
Similar to the problems before it, pick two numbers
that when added together is 75. For this example, it’ll
be 30 and 45
Plug in the degrees into the extended
portion of the formula
Refer back to the Unit circle and
match the coordinates/numbers with
the it corresponding place.
For example:
SIN30: 1
2COS45:
2
2
SIN AND COS DIFFER BY FORMULA BUT ARE SOLVED VERY SIMILARLY
Multiply denominator to denominator,
and numerator to numerator
SIN75°
𝑠𝑖𝑛75
sin 210 − 135
𝑠𝑖𝑛210𝑐𝑜𝑠135 − 𝑐𝑜𝑠210𝑠𝑖𝑛135
(-1
2) × −
2
2− −
3
2×
2
2
2
4− (−
6
4)
Answer:
2+ 6
4
FIND TWO NUMBERS THAT WHEN SUBTRACTED FROM EACH OTHER
EQUALS THE GIVEN NUMBER OF THE PROBLEM. For this example, we
are going to use 210° and 135 which when subtracted equals 75
Plug in the degrees into the formula with the 210
being “α” being 210 and 135 being “β”.
Next, refer back to the Unit Circle and place the
numbers that correspond to what’s being asked for
in the formula. For example:
Sin210 = −1
2cos135 =
− 2
2
SIMPLIFY.
WHY DON’T YOU TAKE A 5 MINUTE BREAK?
tan75° TAN75
tan 45+ tan 30
1 −tan 45 tan 30
1 + 3 3
1− 1 × 3 3
=3+ 3
33 − 3
3
= 3 +3
3 − 3
3 + 3
3 − 3×
3+3
3+3=
3 3 + 3 + 3 3+9
3 − 3 3 + 3 3
12 + 6 3
6= 2 + 3
ANSWER: 2 + 3
For this equation we are first going to use the formula: tan(α + β) tan(𝑎) + tan(𝑏)
1 − tan(𝑎)tan(𝑏)
Breakdown
TAN30 : 1 2
3 2
= 𝟏
𝟑
TAN45: 𝟐𝟐
𝟐𝟐
= 𝟏
SIDE NOTE
• tan =𝑆𝐼𝑁
𝐶𝑂𝑆
•1
3can also bee
written as 3
3
Find two degree that when added together equal the
given degree in the problem For this it would 75, and,
we’re using 45 & 30
Find the tan of the degree (refer to “the
breakdown” section). Next plug in the
number into the formula and find a
common denominator. For this one the common denominator is “3”. So
multiply 1 by 3
3. The denominator 3 will cancel & out
leaving you with 3 +3
3 − 3
Next multiply the denominator by it’s
conjugate. Add together like terms
Next do you basic math solving
(PEMDAS). So, divide 6 in to 12 and
6 3.
tan75
Find the tan of the degree (refer to “the breakdown”
section). Next plug in the number into the formula and find a
common denominator.
For this one the common denominator is “3”. So multiply 1
by 3
3.
The denominator 3 will cancel & out leaving you
with 3 +3
3 − 3
Multiply the whole problem by the
denominator conjugate. Then, add
together like terms
Next do you basic math solving
(PEMDAS). So, divide 6 in to 12
and 6 3.
Breakdown
Tan210: 1 2
− 32
= 3
3
Tan135: 2 2
− 2 2
= −1
We’ll be using the formula: tan(α – β)
WHEN YOU THOUGHT YOU’RE DONE WITH THE PDF ONLY TO FIND OUT
THAT YOU HAVE TO DO THE PRACTICE PROBLEMS AT THE END OF IT
Practice Problems
1.) Solve sin105 using sin(α + β) or sin(α - β)
2.) Solve sin195
3.) Solve cos285
4.) Solve cos-15 using cos(α – β)
5.) Solve tan105 using the formula, tan(α – β)
• SIN105°
sin 60 + 45
𝑠𝑖𝑛60cos45 + cos60sin40
(3
2×
2
2) + (
1
2×
2
2)
or sin 150 − 45
𝑠𝑖𝑛150𝑐𝑜𝑠45 − 𝑐𝑜𝑠150𝑠𝑖𝑛45
(1
2×
2
2) – (-
3
2×
2
2)
Answer:
6 + 2
4
COS285°SIN195° sin 135 + 60
𝑠𝑖𝑛135𝑐𝑜𝑠60 + 𝑐𝑜𝑠135𝑠𝑖𝑛60
(−2
2×
1
2) + (
2
2×
3
2)
or sin 240 − 45
𝑠𝑖𝑛240cos45 - cos240sin45
(-1
2×
2
2) - (-
3
2×
2
2)
Answer:
− 2 + 6
4
cos 315 − 30 𝑐𝑜𝑠315𝑐𝑜𝑠30 − 𝑠𝑖𝑛315𝑠𝑖𝑛30
2
2×
3
2+
2
2×
1
2
Or solve it in this way
cos 240 + 45 𝑐𝑜𝑠240𝑐𝑜𝑠45 − 𝑠𝑖𝑛240𝑠𝑖𝑛45
( -1
2×
2
2) – (-
3
2×
2
2)
Answer:
6 − 2
4
cos(30-45)
𝑐𝑜𝑠30𝑐𝑜𝑠45 − 𝑠𝑖𝑛30𝑠𝑖𝑛45
(3
2×
2
2) + (
1
2×
2
2)
Answer:6 + 2
4
Cos-15° tan105°
Tan(135 – 30)
tan135 − tan301 + (tan135)(tan30)
−1 − 3 3
1+ −1 ( 3 3)
−3 − 3 3
3 +(− 3 3)
−3 − 3
3 − 3×
3+ 3
3+ 3
−9 − 3 3 −3 3 −3
9 − 3 3 + 3 3 − 3
−12 −6 3
6
Answer:
−2 − 3
Tan135: 2 2
− 22
= −1
Tan30: 1 2
3 3
=1
3𝑜𝑟
3
3
For further explanations…
http://www.purplemath.com/modules/idents.html
http://www.shmoop.com/trig-functions/trig-sum-difference-identities.html
http://www.youtube.com/watch?v=SYeBT1xjTa8 (explanation video)
http://www.algebralab.org/lessons/lesson.aspx?file=Trigonometry_TrigSumDif
ference.xml
For further explanation for tan75°:
http://www.youtube.com/watch?v=SYeBT1xjTa8
On page 556 in the “Algebra and Trigonometry: Fifth Element” book