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Here are a few recommended readings before getting started with this lesson.
To be able to solve the tasks of the quest, the class first stopped at the infopoint to recall the Double-Angle Identities. These identities relate the trigonometric values of an angle to the trigonometric values of twice that angle.
The double-angle identities materialize when two angles with the same measure are substituted into the angle sum identities.
Approaching the first equation, the Commutative Property of Multiplication can be applied to the second term of its right-hand side. Then, by adding the terms on the right-hand side of this equation, the formula for sin2θ is obtained.
Approaching the second equation, the Product of Powers Property can be used to rewrite its right-hand side. By doing this, the first identity for the cosine of the double of an angle is obtained.
sin2θ=1−cos2θ
Distribute -1
Add terms
Write as a difference of fractions
Cross out common factors
Cancel out common factors
bmam=(ba)m
ca⋅b=a⋅cb
cos(θ)sin(θ)=tan(θ)
To calculate the exact value of cos120∘, these steps can be followed.
Find the value of sinθ by using one of the Pythagorean Identities.
cosθ=52
(ba)m=bmam
Calculate power
LHS−254=RHS−254
Rewrite 1 as 2525
Subtract fractions
sinθ=521, cosθ=52
Multiply fractions
a⋅cb=ca⋅b
sinθ=31
(ba)m=bmam
Calculate power
LHS−91=RHS−91
Rewrite 1 as 99
Subtract fractions
LHS=RHS
ba=ba
Split into factors
Calculate root
sinθ=31, cosθ=-322
a(-b)=-a⋅b
Multiply fractions
a⋅cb=ca⋅b
sinθ=31, cosθ=-322
(-a)2=a2
(ba)m=bmam
Calculate power
Multiply
Subtract fractions
sin2θ=-942, cos2θ=97
ba/dc=ba⋅cd
Multiply fractions
Cross out common factors
Cancel out common factors
Before moving to the next station, the class stopped at another infopoint to learn about the Half-Angle Identities.
The half-angle identities are special cases of angle difference identities. To evaluate trigonometric functions of half an angle, the following identities can be applied.
The sign of each formula is determined by the quadrant where the angle 2θ lies.
These identities are useful when finding the exact value of the sine, cosine, or tangent at a given angle.
LHS−1=RHS−1
LHS/(-2)=RHS/(-2)
Rearrange equation
Put minus sign in front of fraction
-(b−a)=a−b
LHS=RHS
LHS+1=RHS+1
LHS/2=RHS/2
Rearrange equation
LHS=RHS
sin2θ=±21−cosθ, cos2θ=±21+cosθ
ba=ba
c/da/b=ba⋅cd
Cross out common factors
Cancel out common factors
Multiply fractions
Consider the calculation of the exact value of cos15∘.
θ=30∘
Calculate quotient
1=aa
Subtract fractions
ba/c=b⋅ca
ba=ba
Calculate root
θ=45∘
Calculate quotient
1=aa
Add fractions
ba/c=b⋅ca
ba=ba
Calculate root
Calculate quotient
1=aa
Add and subtract fractions
ba/dc=ba⋅cd
Multiply fractions
ba=b/2a/2
ba=b⋅(2−3)a⋅(2−3)
(a+b)(a−b)=a2−b2
a⋅a=a2
(a−b)2=a2−2ab+b2
Calculate power
Add and subtract terms
1a=a
sinθ=-1715
(-a)2=a2
(ba)m=bmam
Calculate power
LHS−289225=RHS−289225
Rewrite 1 as 289289
Subtract fractions
LHS=RHS
ba=ba
Calculate root
cosθ=178
Rewrite 1 as 1717
Subtract fractions
ba/c=b⋅ca
ba=ba
Calculate root
cosθ=178
Rewrite 1 as 1717
Add fractions
ba/c=b⋅ca
ba=ba
Calculate root
Use the Pythagorean Identity and the Double-Angle Identity for sine.
LHS+sin22x=RHS+sin22x
LHS−4cos4x=RHS−4cos4x
Factor out 4cos2x
1−cos2(θ)=sin2(θ)
sin2(θ)=(sin(θ))2
sin(2θ)=2sin(θ)cos(θ)
(a⋅b)m=am⋅bm
Rewrite one or both sides of the equation by using the Half-Angle Identity or the Double-Angle Identity.
In order to check whether the equation Zosia formed is an identity, rewrite the sides until they match. There are two ways to do this — using the Half-Angle Identity or using the Double-Angle Identity. Each way will be shown one at a time.
Substitute expressions
a⋅b=a⋅b
Multiply fractions
(a−b)(a+b)=a2−b2
1a=1
1−cos2(θ)=sin2(θ)
ba=ba
Calculate root
cos(2θ)=cos2(θ)−sin2(θ)
a2−b2=(a+b)(a−b)
Associative Property of Addition
Cross out common factors
Cancel out common factors
1a=a
Use the Double-Angle Identity for sine and the Tangent Identity.
H=2gv2sin2θ, D=gv2sin2θ
ca⋅b=ca⋅b
c/da/b=ba⋅cd
Multiply fractions
Cross out common factors
Cancel out common factors
sin(2θ)=2sin(θ)cos(θ)
Multiply
Cross out common factors
Cancel out common factors
Write as a product of fractions
tan(θ)=cos(θ)sin(θ)