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| 12 Theory slides |
| 9 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
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)
Find the exact value of the following expressions given that cosθ=1715 and 0∘<θ<90∘.
We are asked to find the value of sin 2θ. To do so, let's start by recalling the Double-Angle Identity for sine. sin 2θ=2sinθcosθ We are given the value of cosθ. However, we do not know the value of sin θ. In order to find it, we can use one of the Pythagorean Identities. sin^2 θ+cos^2 θ=1 Let's substitute cosθ with 1517 and solve the equation for sinθ.
We found that sin θ =± 817. To find the sign of sin θ, recall that θ is greater than 0^(∘) and less than 90^(∘). This means that if we draw the angle θ in standard position, the terminal side will be located in Quadrant I.
We can see that sin θ is positive in Quadrant I. sin θ = 8/17 Now that we know both sinθ and cosθ, we can substitute these two values into the Double-Angle Identity and calculate sin 2θ.
We are asked to find the value of tan 2θ. First, let's recall the Double-Angle Identity for tangent.
tan 2θ=2tan θ/1-tan^2 θ
As we can see, we need to find the value of tanθ. To do so, we can use the Tangent Identity.
tanθ=sinθ/cosθ
Let's substitute 817 for sinθ and 1517 for cosθ into the equation.
Now that the value of tanθ is known, we can calculate tan2θ.
Find the exact value of the following expressions given that tanθ=34 and π<θ<23π.
We are asked to find the value of sin θ2. To do so, let's recall the Half-Angle Identity for sine. sin θ2=± sqrt(1-cosθ/2) First, we need to find the value of cosθ. We know that tanθ= 43, so let's use this piece of information. Recall one of the Pythagorean Identities that involves tangent. 1+tan^2 θ=sec^2 θ There is also a Reciprocal Identity that describes the relationship between secant and cosine. sec θ=1/cosθ By substituting 1cosθ for secθ into the mentioned Pythagorean Identity, an equation that relates tangent and cosine can be obtained. 1+tan^2 θ=( 1/cosθ)^2 Now, we can substitute the known value of tanθ and solve the equation for cosθ.
We found that cos θ =± 35. Now, we need to determine the sign of cos θ. Recall that θ is greater than π and less than 3π2. This means that if we draw the angle θ in standard position, the terminal side will be located in Quadrant III.
We can see that cos θ is negative for θ in Quadrant III. cos θ =- 3/5 We will substitute the value of cosθ into the Half-Angle Identity and calculate sin θ2.
Finally, we need to determine the sign of sin θ2. Knowing that θ is greater than π and less than 3π2, we can find the range of values for θ2. π< θ < 3π/2 ⇔ π2 < θ/2 < 3π/4 Since θ2 is greater than π2 and less than 3π4, it is located in Quadrant II.
Therefore, sin θ2 is positive. sin θ/2= 2sqrt(5)/5
Here, we are asked to find the value of cos θ2. Let's start by recalling the Half-Angle Identity for cosine. cosθ/2=± sqrt(1+cosθ/2) In Part A, we found that cosθ=- 35. Let's substitute this value into the identity and calculate cos θ2.
Previously, we found that θ2 is located in Quadrant II, where the cosine is negative. Therefore, cos θ2 is negative. cos θ/2= - sqrt(5)/5
Use a Double-Angle Identity to find the exact value of each expression.
We are asked to use a Double-Angle Identity to find the exact value of sin 300^(∘). Let's recall the Double-Angle Identity that involves sine. sin 2θ = 2 sin θ cos θ Now, we can use this formula to find the value of sin 300^(∘). We will start by rewriting 300^(∘) as a product where one of the factors is 2.
According to the table of trigonometric values of common angles, sin 150^(∘) = 12 and cos 150^(∘) = - sqrt(3)2. Therefore, we can substitute these two values into our expression.
We want to use a Double-Angle Identity to find the exact value of cos 450^(∘). Let's recall the Double-Angle Identity that involves cosine.
cos 2θ = 2 cos^2 θ - 1
To use this formula, we need to write 450^(∘) as a product where one factor is 2.
By using the table of trigonometric values of notable angles, we can recall that cos 225^(∘) = - sqrt(2)2. Let's substitute this value into our expression and evaluate it.
Use a Half-Angle Identity to find the exact value of each expression.
We are asked to use a Half-Angle Identity to find the exact value of cos 15^(∘). First, let's review the Half-Angle Identity that involves cosine. cos θ/2 = ± sqrt(1+cos θ/2) Next, we can use this formula to find the value of cos 15^(∘). We will start by rewriting 15^(∘) as a quotient of some number and 2.
According to the table of trigonometric values of common angles, cos 30^(∘) = sqrt(3)2. Therefore, we can substitute this value into our expression.
Finally, we need to determine the sign of the expression. To do so, let's recall the signs of sine, cosine, and tangent in the four quadrants of the coordinate plane.
Since 15^(∘) is located in Quadrant I, we know that cos 15^(∘) is positive. cos 15^(∘)= sqrt(2+sqrt(3))/2
We need to find the exact value of tan 22.5^(∘). To do so, let's recall the Half-Angle Identity for tangent. tan θ/2=± sqrt(1-cosθ/1+cosθ) First, we will express 22.5 as the quotient of some number and 2.
From the table of trigonometric values of notable angles, cos 45^(∘) = sqrt(2)2. Therefore, we can substitute this value into our expression.
Finally, we will determine the sign. To do so, let's recall the signs of sine, cosine, and tangent in the four quadrants of the coordinate plane.
Since 22.5^(∘) is located in Quadrant I, we can conclude that tan 22.5^(∘) is positive. tan 22.5^(∘)= sqrt(3 - 2sqrt(2))