2. Sum and Difference Angle Identities
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Chapter 7
2.
Sum and Difference Angle Identities
This lesson delves into the mathematical realm of trigonometric identities, specifically focusing on angle sum and difference identities. It provides a step-by-step guide for calculating trigonometric values using these identities. The use cases are diverse, ranging from academic exercises to practical applications like landscape design and physics. For example, the identities are used to calculate the exact value of sine, cosine, or tangent of a given angle, which can be crucial in fields like engineering and architecture. The lesson also incorporates real-life scenarios, such as calculating the length of a river in a park design, to demonstrate the practical utility of these mathematical tools.
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| | 12 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Sum and Difference Angle Identities
Slide of 12
There are identities that relate the trigonometric values of two angles to the trigonometric values of the sum or difference of these two angles. In this lesson, they will be introduced and practiced.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Designing a City Park
Tiffaniqua, who works as a landscape designer, received a job to create a new design for an old city park. Since the park is quite huge, she divided its area into six rectangular sections. The first section contains a fountain (F) and is crossed by a river at two points — south (S) and north (N).
When she first came to analyze the park, she stood at the north-west corner of the first section, which she marked as point M. Then, she took notes of some measures of angles and distances.
Later when returning to her work space, Tiffaniqua used her notes to make additional calculations. What is the length of the river within the first section of the park? Round the answer to the first decimal place.
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Discussion
Introducing Angle Sum and Difference Identities
There are identities that allow calculating the values of trigonometric functions of the sum or difference of two angles.
Rule
Angle Sum and Difference Identities
To evaluate trigonometric functions of the sum of two angles, the following identities can be applied.
sin(x+y)cos(x+y)tan(x+y)=sinxcosy+cosxsiny=cosxcosy−sinxsiny=1−tanxtanytanx+tany
There are also similar identities for the difference of two angles.
sin(x−y)cos(x−y)tan(x−y)=sinxcosy−cosxsiny=cosxcosy+sinxsiny=1+tanxtanytanx−tany
Proof
Let △AFD be a right triangle with hypotenuse 1 and an acute angle with measure x+y.
sin(x+y)=1DF⇓DF=sin(x+y)
The idea now is to rewrite DF in terms of sinx, siny, cosx, and cosy. To do it, draw a ray so that ∠A is divided into two angles with measures x and y. Let C be a point on this ray such that △ACD and △ABC are right triangles. 
sinx=1DC⇒DC=sinxcosx=1AC⇒AC=cosx
Now consider △ABC. Knowing that AC=cosx, the sine of y can be used to write BC in terms of x and y.
siny=ACBC
▼
Solve for BC
BC=cosxsiny
By the Third Angle Theorem, it is known that ∠GAF≅∠GDC. Therefore, m∠GDC=y.
Since the purpose is to rewrite DF, plot a point E on DF such that EC∥AB. This way a rectangle ECBF is formed. The opposite sides of a rectangle have the same length, so EF and CB are equal. Also, CE⊥DF makes △CED a right triangle.
cosy=sinxDE⇓DE=sinxcosy
Finally, by the Segment Addition Postulate, DF is equal to the sum of DE and EF. All these lengths have been rewritten in terms of the sine and cosine of x and y.
DF=DE+EF⇓sin(x+y)=sinxcosy+cosxsiny
This concludes the proof of the first identity. The other identities can be proven using similar reasoning.
Extra
Calculating sin120∘Consider the following process for calculating the exact value of sin120∘.
- To be able to use the angle sum identities, the angle 120∘ needs to be rewritten as the sum of two angles for which the sine and cosine are known. For example, 120∘ can be rewritten as 90∘+30∘.
- Use the first formula for the angle sum.
- Based on the trigonometric ratios of common angles, it is known that sin90∘=1, sin30∘=21, cos90∘=0, and cos30∘=23.
sin120∘
Rewrite
Rewrite 120∘ as 90∘+30∘
sin(90∘+30∘)
sin(x+y)=sinxcosy+cosxsiny
sin90∘⋅cos30∘+cos90∘⋅sin30∘
SubstituteValues
Substitute values
1⋅23+0⋅21
▼
Simplify
OneMult
1⋅a=a
23+0⋅21
ZeroPropMult
Zero Property of Multiplication
23+0
IdPropAdd
Identity Property of Addition
23
Example
Evaluating Trigonometric Expressions at Random Values
When Tiffaniqua came home from work, she saw that her son Davontay and his friend Zain came up with a game. Davontay assigned numbers 1 through 6 to the trigonometric functions of sine, cosine, and tangent, while Zain assigned numbers 1 through 6 to six angle measures.
a sin105∘
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b tan195∘
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c cos75∘
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d tan(-15∘)
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Hint
a Express 105∘ as the sum or difference of two notable angles.
b Use the Angle Sum and Difference Identities for tangent.
c Note that 75∘ can be expressed as the difference of 120∘ and 45∘.
d Try to simplify the fraction by multiplying both the numerator and denominator by such an expression that would remove a radical from the denominator.
Solution
a The exact value of sin105∘ should be found. Recall that the values of the trigonometric functions of some notable angles are known. Try to represent the angle of 105∘ as the sum or difference of two notable angles.
105∘=60∘+45∘
Now, use the Angle Sum Identity for sine.
sin(x+y)=sinxcosy+cosxsiny
Substitute 60∘ for x and 45∘ for y and simplify.
sin(x+y)=sinxcosy+cosxsiny
SubstituteII
x=60∘, y=45∘
sin(60∘+45∘)=sin60∘cos45∘+cos60∘sin45∘
AddTerms
Add terms
sin105∘=sin60∘cos45∘+cos60∘sin45∘
▼
Substitute values
sin105∘=23cos45∘+cos60∘sin45∘
sin105∘=23⋅22+cos60∘sin45∘
sin105∘=23⋅22+21sin45∘
sin105∘=23⋅22+21⋅22
MultFrac
Multiply fractions
sin105∘=46+42
AddFrac
Add fractions
sin105∘=46+2
b Similarly to Part A, start by expressing 195∘ as the sum or difference of two notable angles.
195∘=240∘−45∘
Next, apply the Angle Difference Identity for tangent.
tan(x−y)=1+tanxtanytanx−tany
Substitute 240∘ for x and 45∘ for y, and then find the value of tan195 by solving the equation.
tan(x−y)=1+tanxtanytanx−tany
SubstituteII
x=240∘, y=45∘
tan(240∘−45∘)=1+tan240∘tan45∘tan240∘−tan45∘
SubTerm
Subtract term
tan195∘=1+tan240∘tan45∘tan240∘−tan45∘
Substitute
tan(240∘)=3
tan195∘=1+3tan45∘3−tan45∘
tan195∘=1+3⋅13−1
MultByOne
a⋅1=a
tan195∘=1+33−1
tan195∘=1+33−1
ExpandFrac
ba=b⋅(1−3)a⋅(1−3)
tan195∘=(1+3)(1−3)(3−1)(1−3)
▼
Simplify right-hand side
ExpandDiffSquares
(a+b)(a−b)=a2−b2
tan195∘=12−(3)2(3−1)(1−3)
MultPar
Multiply parentheses
tan195∘=12−(3)23−(3)2−1+3
CalcPow
Calculate power
tan195∘=1−33−3−1+3
AddSubTerms
Add and subtract terms
tan195∘=-223−4
MoveNegDenomToFrac
Put minus sign in front of fraction
tan195∘=-223−4
SimpQuot
Simplify quotient
tan195∘=-(3−2)
Distr
Distribute -1
tan195∘=-3+2
CommutativePropAdd
Commutative Property of Addition
tan195∘=2−3
c First, note that the angle of 75∘ can be represented as the difference of 120∘ and 45∘. Therefore, the Angle Difference Identity for cosine can be used to calculate the value of cos75∘.
75∘=120∘−45∘
Next, use the known values of the trigonometric functions of notable angles 120∘ and 45∘.
cos(x−y)=cosxcosy+sinxsiny
SubstituteII
x=120∘, y=45∘
cos(120∘−45∘)=cos120∘cos45∘+sin120∘sin45∘
SubTerm
Subtract term
cos(75∘)=cos120∘cos45∘+sin120∘sin45∘
▼
Substitute values
cos75∘=-21cos45∘+sin120∘sin45∘
cos75∘=-21⋅22+sin120∘sin45∘
cos75∘=-21⋅22+23sin45∘
cos75∘=-21⋅22+23⋅22
MultFrac
Multiply fractions
cos75∘=-42+46
CommutativePropAdd
Commutative Property of Addition
cos75∘=46−42
SubFrac
Subtract fractions
cos75∘=46−2
d To find the value of tan(-15∘), first express the angle of -15∘ as the sum or difference of two notable angles.
-15∘=30∘−45∘
Next, use the Angle Difference Identity for tangent. Substitute x with 30∘ and y with 45∘ and solve the equation for tan(-15∘).
tan(x−y)=1+tanxtanytanx−tany
SubstituteII
x=30∘, y=45∘
tan(30∘−45∘)=1+tan30∘tan45∘tan30∘−tan45∘
SubTerm
Subtract term
tan(-15∘)=1+tan30∘tan45∘tan30∘−tan45∘
▼
Simplify right-hand side
tan(-15∘)=1+33tan45∘33−tan45∘
tan(-15∘)=1+33⋅133−1
MultByOne
a⋅1=a
tan(-15∘)=1+3333−1
OneToFrac
Rewrite 1 as 33
tan(-15∘)=33+3333−33
AddFrac
Add fractions
tan(-15∘)=33+333−3
DivFracByFracSL
ba/dc=ba⋅cd
tan(-15∘)=33−3⋅3+33
MultFrac
Multiply fractions
tan(-15∘)=3(3+3)3(3−3)
ReduceFrac
ba=b/3a/3
tan(-15∘)=3+33−3
tan(-15∘)=3+33−3
ExpandFrac
ba=b⋅(3−3)a⋅(3−3)
tan(-15∘)=(3+3)(3−3)(3−3)(3−3)
▼
Simplify right-hand side
ExpandDiffSquares
(a+b)(a−b)=a2−b2
tan(-15∘)=(3)2−32(3−3)(3−3)
ProdToPowTwoFac
a⋅a=a2
tan(-15∘)=(3)2−32(3−3)2
ExpandNegPerfectSquare
(a−b)2=a2−2ab+b2
tan(-15∘)=(3)2−32(3)2−63+32
CalcPow
Calculate power
tan(-15∘)=3−93−63+9
AddSubTerms
Add and subtract terms
tan(-15∘)=-612−63
MoveNegDenomToFrac
Put minus sign in front of fraction
tan(-15∘)=-612−63
SimpQuot
Simplify quotient
tan(-15∘)=-(2−3)
Distr
Distribute -1
tan(-15∘)=-2+3
CommutativePropAdd
Commutative Property of Addition
tan(-15∘)=3−2
Example
Evaluating Trigonometric Functions
In the game that Davontay and Zain created and played, Davontay solved everything correctly. Zain, on the other hand, made one mistake. This was on Zain's mind as they came home, so they decided to practice by evaluating more trigonometric functions.
Find the values of the given expressions along with Zain.
a sin1223π
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b cos(-165∘)
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c tan255∘
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Hint
a The angle 1223π can be expressed as the sum of 1220π and 123π.
b Start by using the Negative Angle Identity for cosine.
c Apply the Angle Sum or Difference Identity for tangent.
Solution
a To calculate the value of sin1223π, start by expressing the angle of 1223π as the sum or difference of two notable angles.
1223π=1220π+123π
Note that the angles 1220π and 123π can be simplified to 35π and 4π, which are the notable angles corresponding to 300∘ and 45∘. Next, use the Angle Sum Identity for sine.
sin(x+y)=sinxcosy+cosxsiny
SubstituteII
x=1220π, y=123π
sin(1220π+123π)=sin1220πcos123π+cos1220πsin123π
AddFrac
Add fractions
sin(1223π)=sin1220πcos123π+cos1220πsin123π
ReduceFrac
ba=b/4a/4
sin(1223π)=sin35πcos123π+cos35πsin123π
ReduceFrac
ba=b/3a/3
sin(1223π)=sin35πcos4π+cos35πsin4π
▼
Substitute values
sin(1223π)=-23cos4π+cos35πsin4π
sin(1223π)=-23⋅22+cos35πsin4π
sin(1223π)=-23⋅22+21sin4π
sin(1223π)=-23⋅22+21⋅22
MultFrac
Multiply fractions
sin(1223π)=-46+42
CommutativePropAdd
Commutative Property of Addition
sin(1223π)=42−46
SubFrac
Subtract fractions
sin(1223π)=42−6
b First, use the Negative Angle Identity for cosine.
cos(-165∘)=cos165∘
Next, note that the angle of 165∘ is the sum of 120∘ and 45∘.
165∘=120∘+45∘
Now, the Angle Sum Identity for cosine can be used.
cos(x+y)=cosxcosy−sinxsiny
SubstituteII
x=120∘, y=45∘
cos(120∘+45∘)=cos120∘cos45∘−sin120∘sin45∘
AddTerms
Add terms
cos165∘=cos120∘cos45∘−sin120∘sin45∘
▼
Substitute values
cos165∘=-21cos45∘−sin120∘sin45∘
cos165∘=-21⋅22−sin120∘sin45∘
cos165∘=-21⋅22−23sin45∘
cos165∘=-21⋅22−23⋅22
MultFrac
Multiply fractions
cos165∘=-42−46
SubFrac
Subtract fractions
cos165∘=4-2−6
FactorOut
Factor out -1
cos165∘=4-(2+6)
MoveNegNumToFrac
Put minus sign in front of fraction
cos165∘=-42+6
c Start by expressing 255∘ as the sum or difference of two notable angles.
255∘=225∘+30∘
Then, use the Angle Sum Identity for tangent.
tan(x+y)=1−tanxtanytanx+tany
SubstituteII
x=225∘, y=30∘
tan(225∘+30∘)=1−tan225∘tan30∘tan225∘+tan30∘
AddTerms
Add terms
tan255∘=1−tan225∘tan30∘tan225∘+tan30∘
▼
Substitute values
tan(225∘)=1
tan255∘=1−1⋅tan30∘1+tan30∘
tan255∘=1−1⋅331+33
IdPropMult
Identity Property of Multiplication
tan255∘=1−331+33
OneToFrac
Rewrite 1 as 33
tan255∘=33−3333+33
AddSubFrac
Add and subtract fractions
tan255∘=33−333+3
DivFracByFracSL
ba/dc=ba⋅cd
tan255∘=33+3⋅3−33
MultFrac
Multiply fractions
tan255∘=3(3−3)(3+3)3
SimpQuot
Simplify quotient
tan255∘=3−33+3
tan255∘=3−33+3
ExpandFrac
ba=b⋅(3+3)a⋅(3+3)
tan255∘=(3−3)(3+3)(3+3)(3+3)
▼
Simplify right-hand side
(a−b)(a+b)=a2−b2
tan255∘=32−(3)2(3+3)(3+3)
MultPow
am⋅an=am+n
tan255∘=32−(3)2(3+3)2
ExpandPosPerfectSquare
(a+b)2=a2+2ab+b2
tan255∘=32−(3)232+23+(3)2
CalcPow
Calculate power
tan255∘=9−39+23+3
AddSubTerms
Add and subtract terms
tan255∘=612+23
ReduceFrac
ba=b/2a/2
tan255∘=36+3
Pop Quiz
Practicing Using the Angle Sum and Difference Identities
Find the trigonometric value of the given angle by using the Angle Sum and Difference Identities. When inputting the answer, write the radical with the greater radicand first.
Example
Comparing Different Answers of an Exercise
The following day at school, Davontay and Zain had a tough test in Physics. After completing it, they discussed how to solve one of the exercises that had them stumped.
Davontay said he wrote cos(60∘+θ) as the answer, however Zain got sin(30∘−θ) as the answer. Are their results equivalent, or did someone made a mistake?
External credits: @brgfx
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Hint
Can cos(60∘+θ) be rewritten into sin(30∘−θ)? Use Cofunction Identities.
Solution
To determine whether the results that Davontay and Zain obtained are equivalent, the following equation should be verified.
This expression is equal to sin(30∘−θ). This means that the expression on the left-hand side of the equation can be rewritten into the expression on the right-hand side. Therefore, Davontay's and Zain's results are equivalent.
cos(60∘+θ)=?sin(30∘−θ)
Use the Angle Sum and Difference Identities for sine and cosine to rewrite both sides of the equation until they match. By one of these identities, the right-hand side can be rewritten as follows.
sin(30∘−θ)=sin30∘cosθ−cos30∘sinθ
Now, try to rewrite the left-hand side to make it match the right-hand side. Note that Cofunction Identities can be useful here.
cos(60∘+θ)
cos(α+β)=cos(α)cos(β)−sin(α)sin(β)
cos60∘cosθ−sin60∘sinθ
Rewrite
Rewrite 60∘ as 90∘−30∘
cos(90∘−30∘)cosθ−sin(90∘−30∘)sinθ
cos(90∘−θ)=sin(θ)
sin30∘cosθ−sin(90∘−30∘)sinθ
sin(90∘−θ)=cos(θ)
sin30∘cosθ−cos30∘sinθ
cos(60∘+θ)=sin(30∘−θ)✓
Example
Checking Whether Two Expressions Are Equivalent
Later, while walking to the cafeteria, Zain and Davontay started jokingly imagining how cool it would be to meet an alien in space. Although they could not go to space themselves — they made weekend plans to build a board game — they came up with an idea to build a small rocket and send their representative Ben!
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Hint
Use the Angle Difference Identity for sine to rewrite Davontay's expression as Zain's expression.
Solution
To determine whether Zain's result is equivalent to Davontay's, the following equation should be verified.
As determined, Zain's expression can be rewritten as Davontay's expression. Therefore, Davontay's and Zain's results are equivalent.
sin(23π−θ)=?-cosθ
Use the Angle Difference Identity for sine to rewrite the left-hand side.
sin(23π−θ)
sin(α−β)=sin(α)cos(β)−cos(α)sin(β)
sin23πcosθ−cos23πsinθ
-1⋅cosθ−cos23πsinθ
-1⋅cosθ−0⋅sinθ
MultPropNegOne
Multiplication Property of -1
-cosθ−0⋅sinθ
ZeroPropMult
Zero Property of Multiplication
-cosθ
sin(23π−θ)=-cosθ✓
Example
Equation of a Standing Wave on a Guitar String
Zain's friend Davontay recently took up guitar lessons. One day, Zain went over to his house to hang out and saw Davontay practicing. Zain told Davontay that they just learned how every time a taut string is pulled and released, a wave is created. Davontay wants to know more!
standing waverepresented by the following formula.
y=Acos(32πt−52πx)+Acos(32πt+52πx)
Here, y is the displacement of each point on the string, which depends on the time t and the point's position x. How can this formula be simplified for t=1? {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI","A"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["-A\\cos((2\\pi x)\/5)"]}}
Hint
Substitute 1 for t and apply the Angle Sum and Difference Identities for cosine.
Solution
The given formula consists of two cosine expressions whose arguments are the sum or difference of the same two angles, namely, 32πt and 52πx. Therefore, use the Angle Sum and Difference Identities for cosine to simplify the formula. Before doing that, however, first substitute 1 for t.
y=Acos(32πt−52πx)+Acos(32πt+52πx)
Substitute
t=1
y=Acos(32π⋅1−52πx)+Acos(32π⋅1+52πx)
MultByOne
a⋅1=a
y=Acos(32π−52πx)+Acos(32π+52πx)
FactorOut
Factor out A
y=A[cos(32π−52πx)+cos(32π+52πx)]
cos(α−β)=cos(α)cos(β)+sin(α)sin(β)
y=A[cos(32π)cos(52πx)+sin(32π)sin(52πx)+cos(32π+52πx)]
cos(α+β)=cos(α)cos(β)−sin(α)sin(β)
y=A[cos(32π)cos(52πx)+sin(32π)sin(52πx)+cos(32π)cos(52πx)−sin(32π)sin(52πx)]
AddSubTerms
Add and subtract terms
y=2Acos(32π)cos(52πx)
y=2A(-21)cos(52πx)
MultPosNeg
a(-b)=-a⋅b
y=-Acos(52πx)
Pop Quiz
Find the Exact Values of Trigonometric Expressions
Consider the given expression involving trigonometric functions. Instead of calculating the value of each trigonometric function, first simplify the expression by applying the Angle Sum and Difference Identities.
Example
Playing a Board Game
After building the rocket and successfully launching it into the neighbour's garden, Zain and Davontay went home and created an amazing board game from scratch! Its central circular part is divided into six equal sectors. 
First, Davontay spun the arrow and placed his rabbit chip at the point where the arrow stopped. Zain then did the same, placing their flamingo chip.
Therefore, the cosine of ∠1 is the cosine of the sum of θ and 60∘.
As can be seen, ∠2 can be expressed as the difference between two central angles and x. This means that the following is true.
a What is the cosine of ∠1?
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b What is the sine of ∠2?
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Hint
a Express the measure of ∠1 as the sum of the central angle's measure and θ. Then use one of the Angle Sum Identities.
b Express the measure of ∠2 as the difference of two central angles' measures and x. Then use one of the Angle Difference Identities.
Solution
a It is given that the circle is divided into six equal sectors. The measure of a full circle is 360∘, which means that the central angle of each sector is 6360=60∘. Note that ∠1 can be expressed as the sum of one central angle of 60∘ and θ.
cosm∠1=cos(θ+60∘)
Now, the Angle Sum Identity for cosine can be used to find the value of cos(θ+60∘).
cos(θ+60∘)
cos(α+β)=cos(α)cos(β)−sin(α)sin(β)
cosθcos60∘−sinθsin60∘
cosθ⋅21−sinθsin60∘
cosθ⋅21−sinθ⋅23
FactorOut
Factor out 21
21(cosθ−sinθ⋅3)
CommutativePropMult
Commutative Property of Multiplication
21(cosθ−3sinθ)
b Start by analyzing ∠2 on the diagram.
sinm∠2=sin(120∘−x)
Now, use the Angle Difference Identity for sine to calculate the value of sinm∠2.
Closure
Calculating the Length of the River
In the challenge at the beginning, it was said that a landscape designer Tiffaniqua got received a job to create a new design for an old city park. She divided its area into six rectangular sections. The first section contains a fountain (F) and is crossed by a river at two points — south (S) and north (N).
When she first came to analyze the park, she stood at the north-west corner of the first section, which she marked as point M. She then took notes of some measures of angles and distances.
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Hint
Start by calculating MF by using the the ratio of the sine of 30∘. Find the measure of the angle the river forms with the right side of the section and then use the Angle Difference Identity.
Solution
To calculate the lengths of the river in the first section, NS should be found. For the purpose of the following calculations, let R be the right upper corner of the rectangular section.
Since the section is a rectangle, ∠NRS is a right angle, which means that △NRS is a right triangle. Additionally, the lengths of the opposite sides of a rectangle are equal, so MF=RS. To find the length of these sides, consider △MSF.
sin30∘=MSMF
Substitute MS with 40 and solve the equation for MF.
sin30∘=MSMF
Substitute
MS=40
sin30∘=40MF
▼
Solve for MF
MultEqn
LHS⋅40=RHS⋅40
40sin30∘=MF
RearrangeEqn
Rearrange equation
MF=40sin30∘
MF=40(21)
Multiply
Multiply
MF=20
30∘+45∘+m∠NSR=90∘⇓m∠NSR=15∘
Now it is known that one of the acute angles in △NSR has measure of 15∘ and its adjacent side is 20 meters long. 
cos15∘=NSSR⇓cos15∘=NS20
Note that 15∘ can be expressed as the difference of 45∘ and 30∘ — notable angles for which trigonometric ratios are known. Therefore, the value of cos15∘ can be calculated by using the Angle Difference Identity for cosine.
cos(x−y)=cosxcosy+sinxsiny
Substitute x with 45∘ and y with 30∘ and find the value of cos15∘.
cos(x−y)=cosxcosy+sinxsiny
SubstituteII
x=45∘, y=30
cos(45∘−30∘)=cos45∘cos30∘+sin45∘sin30∘
SubTerm
Subtract term
cos15∘=cos45∘cos30∘+sin45∘sin30∘
▼
Substitute values
cos15∘=22cos30∘+sin45∘sin30∘
cos15∘=22⋅23+sin45∘sin30∘
cos15∘=22⋅23+22sin30∘
cos15∘=22⋅23+22⋅21
MultFrac
Multiply fractions
cos15∘=46+42
AddFrac
Add fractions
cos15∘=46+2
cos15∘=NS20
▼
Solve for NS
MultEqn
LHS⋅NS=RHS⋅NS
NScos15∘=20
DivEqn
LHS/cos15∘=RHS/cos15∘
NS=cos15∘20
Substitute
cos15∘=46+2
NS=46+220
DivByFracD
b/ca=ba⋅c
NS=6+220⋅4
Multiply
Multiply
NS=6+280
UseCalc
Use a calculator
NS=20.705523…
RoundDec
Round to 1 decimal place(s)
NS≈20.7
Sum and Difference Angle Identities
Find the exact value of each trigonometric expression.
cos(-300∘)
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tan75∘
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cos1211π
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Before we rewrite the given expression, let's start by recalling the values of the three main trigonometric functions for the most important angles.
| sin θ | cos θ | tan θ | |
|---|---|---|---|
| θ =0^(∘) | 0 | 1 | 0 |
| θ =30^(∘) | 1/2 | sqrt(3)/2 | sqrt(3)/3 |
| θ =45^(∘) | sqrt(2)/2 | sqrt(2)/2 | 1 |
| θ =60^(∘) | sqrt(3)/2 | 1/2 | sqrt(3) |
| θ =90^(∘) | 1 | 0 | - |
| θ =180^(∘) | 0 | - 1 | 0 |
| θ =360^(∘) | 0 | 1 | 0 |
Now, let's review the Angle Difference Identity for a cosine function. cos(A-B)=cos A cos B+sin A sin B Note that - 300 can be expressed as the difference of 60 and 360. This means that we can use this identity to rewrite the given expression.
Next, we will use the table at the beginning to find the exact value of the expression. Both 60^(∘) and 360^(∘) appear in the table and, therefore, we can use the values of trigonometric functions for these angles. cos 60^(∘) cos 360^(∘) + sin 60^(∘) sin 360^(∘) = 1/2 (1)+ sqrt(3)/2 (0) Finally, we can evaluate the obtained expression!
Therefore, the value of cos (- 300 ^(∘)) is 12.
Let's start by recalling the Angle Sum Identity for a tangent function. tan(A+B)=tan A + tan B/1-tan A tan B We will use this identity to rewrite the given expression. Note that 75^(∘) can be expressed as the sum of 30^(∘) and 45^(∘), which are two notable angles.
Now, we can substitute the known values of tan 30^(∘) and tan 45^(∘) into the expression. tan 30^(∘) + tan 45^(∘)/1- tan 30^(∘) tan 45^(∘)=sqrt(3)3 + 1/1-( sqrt(3)3) 1 Finally, let's simplify the obtained expression!
Let's also simplify the expression by rationalizing the denominator. To do so, we will multiply the numerator and denominator of the fraction by the conjugate of the denominator. sqrt(3)+3/3-sqrt(3) [0.5em] ⇓ [0.5em] sqrt(3)+3/3-sqrt(3) * 3+sqrt(3)/3+sqrt(3) Let's simplify the expression.
We have found that the value of tan 75 ^(∘) is sqrt(3)+2.
Let's review the values of the three main trigonometric functions for the most important angles one more time.
| sin θ | cos θ | tan θ | |
|---|---|---|---|
| θ =0 | 0 | 1 | 0 |
| θ =π/6 | 1/2 | sqrt(3)/2 | sqrt(3)/3 |
| θ =π/4 | sqrt(2)/2 | sqrt(2)/2 | 1 |
| θ =π/3 | sqrt(3)/2 | 1/2 | sqrt(3) |
| θ =π/2 | 1 | 0 | - |
| θ =π | 0 | - 1 | 0 |
| θ =2π | 0 | 1 | 0 |
We can use the Angle Difference Identity for cosine to rewrite the given expression.
Next, we will use the table we constructed at the beginning of this solution to simplify this expression. cos π cos π/12 - sin π sin π/12 = ( - 1) cos π/12-( 0) sin π/12 Let's simplify the obtained expression!
Therefore, we found that cos 11π12 = - cos π12. Be aware that π12 is the difference of π3 and π4. This means that we can rewrite cos π12 as the cosine of a difference. cos π/12=cos (π/3-π/4) We can once again use the Angle Difference Formula to find the exact value of the expression.
Next, we will use the table at the beginning to substitute the known trigonometric values of the angles appearing in the expression. cos π/3 cos π/4+ sin π/3 sin π/4 = ( 1/2) sqrt(2)/2+( sqrt(3)/2) sqrt(2)/2 Let's finally simplify the obtained expression!
Therefore, cos π12= sqrt(2)+sqrt(6)4. Finally, we can use this information to calculate our original expression.
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Find the value of each trigonometric expression.
cos167∘cos32∘+sin167∘sin32∘
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sin83∘cos7∘+cos83∘sin7∘
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Let's start by recalling the Angle Difference Identity for a cosine function. cos( x- y)= cos xcos y + sin xsin y The given trigonometric expression has the same structure as the expression on the right-side of the identity. Therefore, we can use the identity to evaluate it. cos 167^(∘) cos 32^(∘) +sin 167^(∘) sin 32^(∘) In this case, x= 167^(∘) and y= 32. We will use this formula to simplify and evaluate the given expression.
Now, let's review the Angle Sum Identity for a sine function.
sin( x+ y)=sin xcos y+cos xsin y
Comparing the right-hand side of the identity with the given trigonometric expression, we can see that they have the same structure.
sin 83^(∘)cos 7^(∘)+cos 83^(∘)sin 7^(∘)
Therefore, we have x= 83^(∘) and y= 7^(∘). We can use this formula to simplify and evaluate the given expression.
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Rewrite each expression as a trigonometric function of a single angle measure.
cos9θcos6θ+sin6θsin9θ
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cos11θsin4θ−cos4θsin11θ
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We want to rewrite the given expression as a trigonometric function of a single angle measure. To do so, we will first rearrange the product of sines in the expression by applying the Commutative Property of Multiplication. cos 9θcos 6θ+ sin 6θ sin 9θ ⇕ cos 9θcos 6θ+ sin 9θ sin 6θ Now recall the Angle Difference Identity for cosine. cos (A-B) = cos A cos B + sin A sin B Notice that the given expression is actually equivalent to the cosine of the difference between two angles. Let's use the identity to rewrite our expression!
Similarly to Part A, let's begin by rearranging the factors of the first product in the expression by applying the Commutative Property of Multiplication.
cos 11θ sin 4θ-cos 4θsin 11θ
⇕
sin 4θ cos 11θ-cos 4θsin 11θ
Now, the expression matches the formula for the Angle Difference Identity for sine.
sin (A-B) = sin A cos B - cos A sin B
Therefore, we can use this identity to rewrite our expression.
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Use the sum and difference formulas to check whether each equation is an identity.
sin(23π−θ)=-cosθ
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cos(π+θ)=-sinθ
{"type":"choice","form":{"alts":["Yes","No"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
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We want to determine if the given equation is an identity. sin (3π/2 - θ )? = - cos θ To do so, we can use the Angle Difference Identity for sine. sin (A- B) = sin A cos B - cos A sin B With this relationship in mind, let's rewrite the left-hand side of the given equation and see if it is equivalent to the right-hand side.
After rewriting, the expressions on both side are the same. Therefore, the given equation is an identity meaning that it is true for any value of θ.
Again, we are asked to check whether the following equation is an identity. cos (π + θ)? = cos θ In order to do that, we can use the Angle Sum Identity for cosine. cos (A+ B)=cos A cos B -sin A sin B Let's use it to rewrite the left-hand side of the equation and see if it is equivalent to the right-hand side.
Since the expressions on left- and right-hand sides of the equation are different, it is not an identity.
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Sum and Difference Angle Identities
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