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Trigonometric Identities Sum and Differences

Transcript of Trigonometric Identitiescentralprecalculus.weebly.com/uploads/5/1/5/3/51533951/8_ident_er… ·...

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Trigonometric IdentitiesSum and Differences

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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

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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

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You would need to

frequently refer back to

the Unit Circle while

solving Trigonometric

Identities/ Equations

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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.

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Solve cos75° Cos75°

cos(45+30)

cos45cos30-sin45sin30

(2

3

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

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Sin75°

• Sin75°

• sin(30 + 45)

• 𝑠𝑖𝑛30𝑐𝑜𝑠45 + 𝑐𝑜𝑠30𝑠𝑖𝑛45

•1

2

2+

3

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

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SIN75°

𝑠𝑖𝑛75

sin 210 − 135

𝑠𝑖𝑛210𝑐𝑜𝑠135 − 𝑐𝑜𝑠210𝑠𝑖𝑛135

(-1

2) × −

2

2− −

3

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.

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WHY DON’T YOU TAKE A 5 MINUTE BREAK?

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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.

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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)

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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

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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)

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• SIN105°

sin 60 + 45

𝑠𝑖𝑛60cos45 + cos60sin40

(3

2

2) + (

1

2

2)

or sin 150 − 45

𝑠𝑖𝑛150𝑐𝑜𝑠45 − 𝑐𝑜𝑠150𝑠𝑖𝑛45

(1

2

2) – (-

3

2

2)

Answer:

6 + 2

4

COS285°SIN195° sin 135 + 60

𝑠𝑖𝑛135𝑐𝑜𝑠60 + 𝑐𝑜𝑠135𝑠𝑖𝑛60

(−2

1

2) + (

2

3

2)

or sin 240 − 45

𝑠𝑖𝑛240cos45 - cos240sin45

(-1

2

2) - (-

3

2

2)

Answer:

− 2 + 6

4

cos 315 − 30 𝑐𝑜𝑠315𝑐𝑜𝑠30 − 𝑠𝑖𝑛315𝑠𝑖𝑛30

2

3

2+

2

1

2

Or solve it in this way

cos 240 + 45 𝑐𝑜𝑠240𝑐𝑜𝑠45 − 𝑠𝑖𝑛240𝑠𝑖𝑛45

( -1

2

2) – (-

3

2

2)

Answer:

6 − 2

4

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cos(30-45)

𝑐𝑜𝑠30𝑐𝑜𝑠45 − 𝑠𝑖𝑛30𝑠𝑖𝑛45

(3

2

2) + (

1

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

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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