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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 and is crossed by a river at two points — south and north
When she first came to analyze the park, she stood at the north-west corner of the first section, which she marked as point 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.
There are also similar identities for the difference of two angles.
Proof
Let be a right triangle with hypotenuse and an acute angle with measure
By the Third Angle Theorem, it is known that Therefore,
Since the purpose is to rewrite plot a point on such that This way a rectangle is formed. The opposite sides of a rectangle have the same length, so and are equal. Also, makes a right triangle.
Extra
CalculatingConsider the following process for calculating the exact value of
- To be able to use the angle sum identities, the angle needs to be rewritten as the sum of two angles for which the sine and cosine are known. For example, can be rewritten as
- Use the first formula for the angle sum.
- Based on the trigonometric ratios of common angles, it is known that and
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 through to the trigonometric functions of sine, cosine, and tangent, while Zain assigned numbers through to six angle measures.
a
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b
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c
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d
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Hint
a Express as the sum or difference of two notable angles.
b Use the Angle Sum and Difference Identities for tangent.
c Note that can be expressed as the difference of and
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 should be found. Recall that the values of the trigonometric functions of some notable angles are known. Try to represent the angle of as the sum or difference of two notable angles.
b Similarly to Part A, start by expressing as the sum or difference of two notable angles.
▼
Simplify right-hand side
MultPar
Multiply parentheses
CalcPow
Calculate power
AddSubTerms
Add and subtract terms
MoveNegDenomToFrac
Put minus sign in front of fraction
SimpQuot
Simplify quotient
Distr
Distribute
CommutativePropAdd
Commutative Property of Addition
c First, note that the angle of can be represented as the difference of and Therefore, the Angle Difference Identity for cosine can be used to calculate the value of
SubTerm
Subtract term
▼
Substitute values
MultFrac
Multiply fractions
CommutativePropAdd
Commutative Property of Addition
SubFrac
Subtract fractions
d To find the value of first express the angle of as the sum or difference of two notable angles.
SubTerm
Subtract term
▼
Simplify right-hand side
CalcPow
Calculate power
AddSubTerms
Add and subtract terms
MoveNegDenomToFrac
Put minus sign in front of fraction
SimpQuot
Simplify quotient
Distr
Distribute
CommutativePropAdd
Commutative Property of Addition
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
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b
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c
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Hint
a The angle can be expressed as the sum of and
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 start by expressing the angle of as the sum or difference of two notable angles.
AddFrac
Add fractions
▼
Substitute values
MultFrac
Multiply fractions
CommutativePropAdd
Commutative Property of Addition
SubFrac
Subtract fractions
b First, use the Negative Angle Identity for cosine.
AddTerms
Add terms
▼
Substitute values
MultFrac
Multiply fractions
SubFrac
Subtract fractions
FactorOut
Factor out
MoveNegNumToFrac
Put minus sign in front of fraction
c Start by expressing as the sum or difference of two notable angles.
AddTerms
Add terms
▼
Substitute values
IdPropMult
Identity Property of Multiplication
OneToFrac
Rewrite as
AddSubFrac
Add and subtract fractions
MultFrac
Multiply fractions
SimpQuot
Simplify quotient
▼
Simplify right-hand side
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.
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Davontay said he wrote as the answer, however Zain got as the answer. Are their results equivalent, or did someone made a mistake? {"type":"choice","form":{"alts":["Someone made a mistake.","Their answers are equivalent."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}
Hint
Can be rewritten into Use Cofunction Identities.
Solution
To determine whether the results that Davontay and Zain obtained are equivalent, the following equation should be verified.
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.
Now, try to rewrite the left-hand side to make it match the right-hand side. Note that Cofunction Identities can be useful here.
This expression is equal to 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.
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!
{"type":"choice","form":{"alts":["Yes","No"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
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.
Use the Angle Difference Identity for sine to rewrite the left-hand side.
As determined, Zain's expression can be rewritten as Davontay's expression. Therefore, Davontay's and Zain's results are equivalent.
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.
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Hint
Substitute for 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, and Therefore, use the Angle Sum and Difference Identities for cosine to simplify the formula. Before doing that, however, first substitute for
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 is the cosine of the sum of and
Now, the Angle Sum Identity for cosine can be used to find the value of
As can be seen, can be expressed as the difference between two central angles and This means that the following is true.
Now, use the Angle Difference Identity for sine to calculate the value of
a What is the cosine of
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b What is the sine of
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Hint
a Express the measure of as the sum of the central angle's measure and Then use one of the Angle Sum Identities.
b Express the measure of as the difference of two central angles' measures and 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 which means that the central angle of each sector is Note that can be expressed as the sum of one central angle of and
b Start by analyzing on the diagram.
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 and is crossed by a river at two points — south and north
When she first came to analyze the park, she stood at the north-west corner of the first section, which she marked as point She then took notes of some measures of angles and distances.
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Hint
Start by calculating by using the the ratio of the sine of 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, should be found. For the purpose of the following calculations, let be the right upper corner of the rectangular section.
Since the section is a rectangle, is a right angle, which means that is a right triangle. Additionally, the lengths of the opposite sides of a rectangle are equal, so To find the length of these sides, consider
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