6. Radian Measure and Arc Length
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Chapter 5
6.
Radian Measure and Arc Length
| Student Learning Objectives: |
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Why
Understanding radian measure helps connect angles, circles, and arc lengths in a more flexible way. This lesson introduces radians as another way to measure angles and shows how to convert between radians and degrees. It also explains how radian measure relates directly to arc length, leading to formulas that can be used when the angle is given in radians or degrees. By learning these relationships, students can solve problems involving circles, clocks, Ferris wheels, and orbital motion more efficiently.
Lesson Settings & Tools
| | 12 Theory slides |
| | 12 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Radian Measure and Arc Length
Slide of 12
Angles can be measured not only by degrees but also by radians. In this lesson, the concept of radian measure will be introduced. Additionally, the formulas related to arc length will be derived and exercised.
Here are a few recommended readings before getting started with this lesson.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Try your knowledge on these topics.
a Calculate the circumference of the given circle.
Round the answer to the closest integer.
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b Find the measure of arc AB.
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c Determine the measures of ∠ SOR and ∠ STR.
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Challenge
Investigating the Orbit of a Satellite
In 2007, a satellite named SELENE was launched to explore the Moon's surface for a few years. It orbited the Moon in a circular path staying at a constant altitude of 100 kilometers. Given that the radius of the Moon is 1737 kilometers, find the distance that the satellite traveled when it completed 60 % of its orbit.
External credits: @wirestock
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Discussion
Radian
In order to introduce the concept of radian measure, the definition of a radian should first be explored.
A radian, like a degree, is an angle unit. One radian is defined as the measure of the central angle that intercepts an arc equal in length to the radius of the circle. It corresponds to roughly 57.3^(∘).
radis seldom written. Instead, no unit marker indicates radians. Consider two expressions. cos 64^(∘) and cos 5 The first angle is given in degrees, and the other is given in radians. At first glance, radians might seem inconvenient, but they make calculations simpler in certain circumstances,. Radians are also the SI unit for angles.
Rule
Relationship Between Radians and Degrees
Radians and degrees are two different units of measure for an angle. Knowing how to convert between the two is majestic, especially in trigonometry. Recall that a full circle measures 360^(∘), which corresponds to 2π radians. 360^(∘) = 2 π rad ⇔ 180^(∘) = π rad Using this equality, it is possible to find equivalent expressions for 1^(∘) and 1rad.
| Degrees to Radians | Radians to Degrees |
|---|---|
| 1^(∘) = π/180rad | 1rad = 180^(∘)/π |
Proof
The arc length of one revolution around a circle is the circumference of the circle 2π r. The number of radians in a full revolution is calculated by dividing 2π r by the radius r of the circle.
2π r/r = 2π
This means that one revolution is 2π radians which in degrees is 360^(∘), and that leads to the expression 360^(∘) = 2π rad. Both sides then divided by 2 gives the following equation.
180^(∘) = π rad
Therefore, 180^(∘) correspond to π radians.
From the relation 180^(∘) = π rad, it is possible to find two rules by dividing both sides by either 180 or π. To find an expression for 1^(∘), divide both sides by 180.
180 ^(∘) = π rad
DivEqn
.LHS /180.=.RHS /180.
180/180 ^(∘) = π/180 rad
CalcQuot
Calculate quotient
1 ^(∘) = π/180 rad
As shown, 1^(∘) corresponds to π180rad, which is approximately 0.017 radians. To get an expression for 1rad, divide both sides by π.
180 ^(∘) = π rad
DivEqn
.LHS /π.=.RHS /π.
180^(∘)/π = π/π rad
CalcQuot
Calculate quotient
180^(∘)/π = 1 rad
RearrangeEqn
Rearrange equation
1 rad = 180^(∘)/π
Therefore, 1 radian corresponds to 180^(∘)π, which is approximately 57.3^(∘).
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Pop Quiz
Practice Converting Degrees Into Radians
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Pop Quiz
Practice Converting Radians Into Degrees
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Explore
Investigating the Radian Measure for Different Circles
Consider a circle with a radius of 3 units. Move the point on the circle and pay close attention to the ratio of the arc length to the radius as the arc measure changes.
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Discussion
Comparing Radian Measures of Concentric Circles
To understand the observed relation, consider two concentric circles with different radii r_1 and r_2.
These two circles are similar, as all circles are similar under dilation. Therefore, the ratios of corresponding parts of the circles are proportional. This means the ratio of the radii is equal to the ratio of the arc lengths intercepted by the same central angle. r_1/r_2=s_1/s_2 This proportion can be rewritten as an equivalent equation.
r_1/r_2=s_1/s_2
▼
Rearrange equation
MultEqn
LHS * r_2s_2=RHS* r_2s_2
r_1/r_2* r_2s_2=s_1/s_2 * r_2s_2
CancelCommonFac
Cancel out common factors
r_1/r_2* r_2s_2=s_1/s_2 * r_2s_2
SimpQuotProd
Simplify quotient and product
r_1s_2=s_1r_2
DivEqn
.LHS /r_1.=.RHS /r_1.
s_2=s_1r_2/r_1
MovePartNumRight
a* b/c=a/c* b
s_2=s_1/r_1r_2
DivEqn
.LHS /r_2.=.RHS /r_2.
s_2/r_2=s_1/r_1
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Discussion
Using Radians to Calculate Arc Lengths
The radian measure of a central angle is defined as the quotient between the length of the arc intercepted by the angle and the radius of the circle.
Based on the diagram above, the radian measure of the central angle ∠ O is defined as follows.
θ=s/r
By using this definition, the formula for the length of an arc can be derived.
s=θ r
Here, θ is the angle measure given in radians. However, what if the measure θ of a central angle is given in degrees? In that case, by multiplying θ by the conversion factor π180^(∘), the measure can be converted into radians and substituted into the formula. s=( π/180^(∘)θ) r [0.3cm] ⇕ s=π r(θ/180^(∘)) This formula is often written in the following equivalent manner.
s=2π r(θ/360^(∘))
This equivalent form is convenient to work with because 2π r is the circumference of a circle. Since a full circle measures 360^(∘), dividing 2π r by 360 results in the length of an arc intercepted by a 1-degree central angle. Finally, by multiplying that value by the measure of the arc, its length is obtained.
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Example
Arc Length Between the Long and Short hand of a Clock
In her room Paulina has a big clock whose longer hand is 9 centimeters long. When Paulina came home and looked at her clock, it was exactly 2P.M. She calculated the angle between the longer and shorter hands of the clock to be 60^(∘).
After a Geometry lesson, Paulina started wondering how long the arc formed by the hands of the clock is. Based on the length of the longer hand, she estimates the radius of the clock to be 15 centimeters. Help Paulina calculate the length of the arc. Round the answer to one decimal place.
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Hint
Since the central angle is given in degrees, the arc length can be calculated using the formula s=2π r( θ360^(∘)).
Solution
To find the length of an arc using the measure of the corresponding central angle given in degrees, the following formula can be used.
s=2π r (θ/360^(∘))
It is known that the radius of the clock is about 15 centimeters and the central angle that intercepts the arc measures 60^(∘).
By substituting these values into the above formula, the length of the arc can be determined.
s=2π r (θ/360^(∘))
SubstituteII
r= 15, θ= 60^(∘)
s=2π ( 15) (60^(∘)/360^(∘))
▼
Evaluate right-hand side
ReduceFrac
a/b=.a /60^(∘)./.b /60^(∘).
s=2π (15) (1/6)
MoveLeftFacToNumOne
a* 1/b= a/b
s=2π(15)/6
Multiply
Multiply
s=30π/3
ReduceFrac
a/b=.a /3./.b /3.
s=10π
UseCalc
Use a calculator
s=31.415926...
RoundDec
Round to 1 decimal place(s)
s≈ 31.4
The arc between the hands of the clock is about 31.4 centimeters long.
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Example
Finding the Length of an Arc
After having dinner, Paulina decided to do her math homework. She is given a circle with radius 5 inches and an inscribed angle that measures 40^(∘).
Paulina is asked to convert the measure of the given angle into radians and then use it to find the length of MN. Help Paulina find the correct answer. The length should be rounded to the closest integer.
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Hint
To convert the measure of an angle from degrees to radians, use the conversion factor π180^(∘). The length of the arc can be found by using the formula s=θ r, where θ is the measure of the corresponding central angle in radians.
Solution
It can be seen in the diagram that ∠ MKN, which is an inscribed angle, measures 40^(∘).
By multiplying this value by the conversion factor π180^(∘), the measure of this angle can be converted from degrees to radians.
m∠ MKN = 40^(∘) * π/180^(∘)
MoveLeftFacToNum
a*b/c= a* b/c
m∠ MKN = 40^(∘) π/180^(∘)
ReduceFrac
a/b=.a /20^(∘)./.b /20^(∘).
m∠ MKN = 2π/9
Next, to find the length of MN, the measure of the corresponding central angle should be known. Recall that the measure of the inscribed angle is half the measure of the corresponding central angle. In this case, ∠ MKN corresponds to a central angle ∠ MON.
Using this information, the measure of ∠ MON can be found.
m∠ MKN=1/2m∠ MON
Substitute
m∠ MKN= 2π/9
2π/9=1/2m∠ MON
▼
Solve for m∠ MON
MultEqn
LHS * 2=RHS* 2
2π/9(2)=m∠ MON
MoveRightFacToNum
a/c* b = a* b/c
2π(2)/9=m∠ MON
Multiply
Multiply
4π/9=m∠ MON
RearrangeEqn
Rearrange equation
m∠ MON=4π/9
Finally, the length of MN can be calculated using the corresponding formula. s=θ r ⇓ MN=m∠ MON r Here, 4π9 and 5 can be substituted for m∠ MON and r, respectively.
MN=m∠ MON r
SubstituteII
m∠ MON= 4π/9, r= 5
MN=( 4π/9) ( 5)
▼
Evaluate right-hand side
MoveRightFacToNum
a/c* b = a* b/c
MN=4π(5)/9
Multiply
Multiply
MN=20π/9
UseCalc
Use a calculator
MN=6.981317...
RoundInt
Round to nearest integer
MN≈ 7
The length of MN is approximately 7 inches.
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Example
Finding Angle Measures Given an Arc Length
Finally, Paulina finished all her homework. She could now go to an amusement park with her friends. They get on a Ferris Wheel feeling super excited. The diagram indicates their positions.
External credits: @wirestock
If the length of an arc between each cabin is 3 meters and the radius of the Ferris wheel is 12 meters, what is the measure of the angle formed by Paulina, the center of rotation, and Tiffaniqua? What is the measure of an angle formed by Paulina, Ali, and Tiffaniqua?
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Hint
Identify the type of each angle. Use the formula for the arc length.
Solution
It is given that the length of an arc between each cabin is 3 meters. There is one cabin between Paulina and Tiffaniqua, so the arc between their cabins is 6 meters long.
The length s of an arc of a circle with radius r can be calculated by using the following formula. s=2π r (θ/360^(∘)) Here, θ is the measure of the central angle, in degrees, that intercepts the arc. In this case, this is the angle formed by Paulina, the center of rotation, and Tiffaniqua. By substituting s= 6 and r= 12, its measure can be calculated.
s=2π r (θ/360^(∘))
SubstituteII
s= 6, r= 12
6=2π ( 12) (θ/360^(∘))
▼
Solve for θ
Multiply
Multiply
6=24π(θ/360^(∘))
MoveLeftFacToNum
a*b/c= a* b/c
6=24πθ/360^(∘)
ReduceFrac
a/b=.a /24./.b /24.
6=πθ/15^(∘)
MultEqn
LHS * 15^(∘)=RHS* 15^(∘)
90^(∘)=πθ
RearrangeEqn
Rearrange equation
πθ=90^(∘)
DivEqn
.LHS /π.=.RHS /π.
θ=90^(∘)/π
UseCalc
Use a calculator
θ = 28.647889...^(∘)
RoundDec
Round to 1 decimal place(s)
θ ≈ 28.6^(∘)
The measure of the angle formed between Paulina, the center of the Ferris wheel, and Tiffaniqua is about 28.6^(∘). This information can be used to find the measure of the angle formed by Paulina, Ali, and Tiffaniqua.
This angle, labeled as ∠ 1, is an inscribed angle that intercepts the arc between Paulina and Tiffaniqua. The same arc is intercepted by the central angle found earlier. Therefore, the measure of ∠ 1 is half the measure of that central angle. m∠ 1=28.6^(∘)/2 ⇕ m∠ 1=14.3^(∘) The measure of the angle formed by Paulina, Ali, and Tiffaniqua is about 14.3^(∘).
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Closure
Finding the Orbit of a Satellite
The challenge presented at the beginning of this lesson can now be solved thanks to the concepts covered.
In 2007, a satellite named SELENE was launched to explore the Moon's surface for a few years. It orbited the Moon in a circular path staying at a constant altitude of 100 kilometers. Given that the radius of the Moon is 1737 kilometers, find the distance that the satellite traveled when it completed 60 % of its orbit.
External credits: @wirestock
Round the answer to the closest integer.
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Solution
A full orbit around the moon is the equivalent to one full circle, which has
a measure of 2π radians. By multiplying this value by 0.6, the measure of the arc that corresponds to 60 % of a circumference can be found.
0.6* 2π =1.2π
This relationship can be illustrated in the diagram.
External credits: @wirestock
By the Segment Addition Postulate, the radius of the orbit is equal to the sum of the Moon's radius, which is 1737 kilometers, and 100 kilometers. r=1737+100 ⇔ r=1837km To find the length of the desired arc, the following formula can be used. s=θ r Here, θ is the measure, in radians, of the corresponding central angle. Recall that the measure of an arc is equal to the measure of its corresponding central angle. Therefore, θ equals 1.2π. By substituting θ= 1.2π and r= 1837 into the formula, the value of s can be calculated.
s=θ r
SubstituteII
θ= 1.2π, r= 1837
s=( 1.2π)( 1837)
UseCalc
Use a calculator
s=6925.326845...
RoundInt
Round to nearest integer
s≈ 6925
It has been determined that the distance traveled by the satellite when it completed 60 % of its orbit is 6925 kilometers.
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Radian Measure and Arc Length
Exercise 3.1
Two points, P and Q, are on the circumference of the unit circle in the first quadrant. They have x-coordinates of 0.1 and 0.6, respectively. How many units is the shortest arc we can draw between P and Q? Round the answer to two decimal places.
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Let's start by drawing the unit circle and mark the shortest arc that is created by P and Q on the circle's circumference.
To determine the length of this arc, we will use the following formula. s = θ r In this formula, θ is the measure of the central angle in radians. Since the unit circle has a radius of r= 1, the length of the arc is the same as the central angle. s = θ ( 1) ⇔ s = θ We can calculate θ by subtracting θ_Q from θ_P.
Remember that the x-coordinate of a point on the unit circle is the cosine of the central angle. Therefore, P has a cosine value of 0.1 and Q has a cosine value of 0.6. That allows us to write two equations. cos θ_P &= 0.1 ⇔ θ_P = cos^(-1) 0.1 cos θ_Q &= 0.6 ⇔ θ_Q = cos^(-1) 0.6 Let's substitute these identities in the equation describing θ.
To calculate this we need a graphing calculator. Before we can calculate it though, we must make sure the calculator interprets the argument of trigonometric functions as radians. Push MODE and select Radian
on the third row.
Now, we can continue calculating the measure of θ.
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Solution
Which is longer, the sum of the marked arcs or the perimeter of the pentagon?
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Since we are not given any lengths, we must find expressions which can be compared. Notice that we are comparing a sum of circle radii to the length of the arcs. These can be compared, even without actual lengths, if we work in radians.
Perimeter of Pentagon
Let's label the radius of each circles a.
As we can see, the perimeter of the pentagon equals the sum of two radii for each of the five circles. With this information, we can determine an expression for the perimeter. 5( 2a) = 10 a
Length of Arcs
To calculate the length of the arcs, we need to find the sum of the angle measures of a pentagon. (n-2)* 180^(∘) ⇓ (5-2)* 180^(∘)=540^(∘) The measures of the interior angles sum to 540^(∘). Since the pentagon is regular, each angle has a measure of one fifth of this. 540^(∘)/5=108^(∘) Let's rewrite this from degrees to radians.
Each angle has a measure of 3π5 radians. Notice that each of these angles also is a central angle in a circle.
When we know the measure of each central angle, we can find the arc length using the following formula. s = θ r In this formula, θ is the measure of the central angle in radians.
The length of one arc is 3aπ5. If we multiply this by 5 we get the combined length of the five arcs. 5s = 5(3aπ/5) ⇓ 5s ≈ 9.424a Since 9.424a<10a, we know that the sum of the arc lengths is less than the perimeter of the pentagon.
Here, we are now dealing with a polygon formed by six rather than five circles. Since the polygon will then have six sides, it is a regular hexagon. Let's draw the figure to aid our solving process.
Again, if we label the radius of each circle as a, we get a total perimeter of 6 times 2a. Perimeter of Hexagon: 6(2a)=12a Now we want to calculate the length of the arcs. A hexagon has a total angle sum of 720^(∘) and each of the six angles measures 120^(∘). Let's rewrite this as radians.
Each angle has a measure of 2π3 radians.
Let's calculate the length of one arc and then multiply this by 6 to account for all of the arcs.
Since 12.57a>12a, we know that the sum of the arc lengths is greater than the perimeter of the hexagon.
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Solution
Radian Measure and Arc Length
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