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| 12 Theory slides |
| 12 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.
Try your knowledge on these topics.
Round the answer to the closest integer.
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.
In order to introduce the concept of radian measure, the definition of a radian should first be explored.
radis seldom written. Instead, no unit marker indicates radians. Consider two expressions.
Degrees to Radians | Radians to Degrees |
---|---|
1∘=180π rad | 1 rad=π180∘ |
LHS/π=RHS/π
Calculate quotient
Rearrange equation
To understand the observed relation, consider two concentric circles with different radii r1 and r2.
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.LHS⋅r2s2=RHS⋅r2s2
Cancel out common factors
Simplify quotient and product
LHS/r1=RHS/r1
ca⋅b=ca⋅b
LHS/r2=RHS/r2
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=rs
By using this definition, the formula for the length of an arc can be derived.
s=θr
s=2πr(360∘θ)
This equivalent form is convenient to work with because 2πr is the circumference of a circle. Since a full circle measures 360∘, dividing 2πr 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.
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 2 P.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.Since the central angle is given in degrees, the arc length can be calculated using the formula s=2πr(360∘θ).
r=15, θ=60∘
ba=b/60∘a/60∘
a⋅b1=ba
Multiply
ba=b/3a/3
Use a calculator
Round to 1 decimal place(s)
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.To convert the measure of an angle from degrees to radians, use the conversion factor 180∘π. The length of the arc can be found by using the formula s=θr, where θ is the measure of the corresponding central angle in radians.
m∠MKN=92π
LHS⋅2=RHS⋅2
ca⋅b=ca⋅b
Multiply
Rearrange equation
m∠MON=94π, r=5
ca⋅b=ca⋅b
Multiply
Use a calculator
Round to nearest integer
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.
Identify the type of each angle. Use the formula for the arc length.
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=6, r=12
Multiply
a⋅cb=ca⋅b
ba=b/24a/24
LHS⋅15∘=RHS⋅15∘
Rearrange equation
LHS/π=RHS/π
Use a calculator
Round to 1 decimal place(s)
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.θ=1.2π, r=1837
Use a calculator
Round to nearest integer
Convert the angle measurement from degrees to radians. Give the answer in exact form.
To convert from degrees to radians, we use the following relationship. 180^(∘)=π rad By dividing both sides of this equation by 180^(∘), we get a relationship showing how many radians that corresponds to 1^(∘). 1^(∘)=π/180 rad Next, we can convert the given number of degrees to radians if we multiply both sides by 30.
We will use the same relationship as in Part A to convert 15^(∘) to radians.
Again, let's repeat the procedure from Part A and B.
Like in previous parts, we will repeat the established procedure to convert 450^(∘) to radians.
Convert the angle measurement from radians to degrees.
To convert from radians to degrees, we can use the following formula. π rad=180^(∘) By dividing both sides by π , we get a formula showing the number of degrees that correspond to 1 radian. 1 rad=180^(∘)/π Next, we can convert the angle measure from radians to degrees by multiplying both sides of this relationship by π4.
Let's use the same relationship as in Part A to convert π3 to degrees.
We will repeat the procedure from Part A and B.
Again, we will repeat the established procedure to convert π radians to degrees.
Zain and Tearrik both use their calculators to determine the sine value of 30∘. Wait a minute. They get different results.
Tearrik has the correct answer. The error Zain made is that he set his calculator to use radians instead of degrees. If we determine sin 30 with the calculator set to radians, we would obtain - 0.9880316241. That is the answer which Zain calculated.
To change from radians to degrees, push MODE and then select Degree on the third row.
What is the ratio of the length of the arc to the radius of the circle. Answer in exact form.
What is the ratio of the length of the arc to the radius of the circle. Answer in exact form.
To find the arc length of AB we have to multiply the circumference of the circle with the ratio of the central angle to 360^(∘).
The arc length is 2π3. Finally, we will divide the arc length with the radius of D which is 2. .2π /3./2=2π/2(3)=π/3
As in Part A, we will find the arc length by multiplying the circumference of the circle with the ratio of the central angle to 360^(∘)
The arc length is 4π3. As in Part A, we will divide the arc length with the radius of D. .4π /3./4=4π/4(3)=π/3
To show that the radian measure of a sector that spans 60^(∘) always is π3 we will calculate the sector length for a circle with a radius of x units.
Let's calculate the arc length.
The arc length is xπ3. As in Part A, we will divide the arc length with the radius of ⊙ C. .π x /3./x=π x/3x=π/3 As we can see, the ratio is π3 regardless of the radius of the circle.
The angle measure in radian is the arc length divided by the radius of a circle. Radian= Arc length/Radius If we increase the central angle from 90^(∘) to 180^(∘), we have doubled the arc length. This must also mean that the radian measure of the central angle has doubled. 2(π/2)=π Therefore, the radian measure will equal π.
Calculate the value of the trigonometric expression where the argument is given in radians. Round the value to three decimal places.
The value of sin 4 can not be expressed exactly. This means we must use a calculator to determine this. First, make sure that the calculator is set to radians. Push MODE and select Radian
on the third row.
The calculator is now set to interpret the argument of a trigonometric expressions in radians. If we write sin 4 on the calculator, the result will be the sine value of 4 radians.
When we round to three decimal places, we see that sin 4 has a value of - 0.757.
As in Part A, we need a calculator to determine sin 10. Since the calculator is already set to radians, we can immediately determine the value of sin 10.
When we round to three decimal places, sin 10 has a value of - 0.544.