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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

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

There are identities that allow calculating the values of trigonometric functions of the sum or difference of two angles.

Rule

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 $

Let $△AFD$ be a right triangle with hypotenuse $1$ and an acute angle with measure $x+y.$

By definition, the sine of an angle is the ratio between the lengths of the opposite side and the hypotenuse.$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.
Consider $△ACD.$ By calculating the sine and cosine of $x,$ the legs of this triangle can be rewritten.
$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 $BC$

Substitute

$AC=cosx$

$siny=cosxBC $

MultEqn

$LHS⋅cosx=RHS⋅cosx$

$cosxsiny=BC$

RearrangeEqn

Rearrange equation

$BC=cosxsiny$

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.

Consequently, $EF=cosxsiny$ and $DE$ can be written in terms of $sinx$ and $cosy$ using the cosine ratio.$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.
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

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.

Next, they rolled the dice four times to identify which two trigonometric values each person should calculate. The die on the left determines the trigonometric function and the die on the right determines the angle measure.
Find the exact values of the expressions that Davontay and Zain obtained. Write the answers in such a way that there are no radicals in the denominators.

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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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.

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

$sin105_{∘}=23 cos45_{∘}+cos60_{∘}sin45_{∘}$

$sin105_{∘}=23 ⋅22 +cos60_{∘}sin45_{∘}$

$sin105_{∘}=23 ⋅22 +21 sin45_{∘}$

$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+3 tan45_{∘}3 −tan45_{∘} $

$tan195_{∘}=1+3 ⋅13 −1 $

MultByOne

$a⋅1=a$

$tan195_{∘}=1+3 3 −1 $

$tan195_{∘}=1+3 3 −1 $

ExpandFrac

$ba =b⋅(1−3 )a⋅(1−3 ) $

$tan195_{∘}=(1+3 )(1−3 )(3 −1)(1−3 ) $

▼

Simplify right-hand side

ExpandDiffSquares

$(a+b)(a−b)=a_{2}−b_{2}$

$tan195_{∘}=1_{2}−(3 )_{2}(3 −1)(1−3 ) $

MultPar

Multiply parentheses

$tan195_{∘}=1_{2}−(3 )_{2}3 −(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

$cos75_{∘}=-21 cos45_{∘}+sin120_{∘}sin45_{∘}$

$cos75_{∘}=-21 ⋅22 +sin120_{∘}sin45_{∘}$

$cos75_{∘}=-21 ⋅22 +23 sin45_{∘}$

$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+33 tan45_{∘}33 −tan45_{∘} $

$tan(-15_{∘})=1+33 ⋅133 −1 $

MultByOne

$a⋅1=a$

$tan(-15_{∘})=1+33 33 −1 $

OneToFrac

Rewrite $1$ as $33 $

$tan(-15_{∘})=33 +33 33 −33 $

AddFrac

Add fractions

$tan(-15_{∘})=33 +3 33 −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

$(a+b)(a−b)=a_{2}−b_{2}$

$tan(-15_{∘})=(3 )_{2}−3_{2}(3 −3)(3 −3) $

ProdToPowTwoFac

$a⋅a=a_{2}$

$tan(-15_{∘})=(3 )_{2}−3_{2}(3 −3)_{2} $

ExpandNegPerfectSquare

$(a−b)_{2}=a_{2}−2ab+b_{2}$

$tan(-15_{∘})=(3 )_{2}−3_{2}(3 )_{2}−63 +3_{2} $

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

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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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.

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.