Level 2 - Radian Measure and Arc Length - Circles With and Without Coordinates (Geometry)

Degrees to Radians	Radians to Degrees
$1^{\circ} = \frac{π}{1 8 0} rad$	$1 rad = \frac{1 8 0 ^{\circ}}{π}$

Degrees to Radians

Radians to Degrees

1^{\circ} = \frac{π}{1 8 0} rad

1 rad = \frac{1 8 0 ^{\circ}}{π}

Use the unit circle to find the value of the following trigonometric expression where the argument is given in radians. Give the answer in exact form.

sin \frac{π}{4}

sin \frac{4 π}{3}

To determine the value of sin π4, we will first convert the angle measure from radians to degrees. To do that, we will use the following formula. 180^(∘)=π By dividing this formula by 4, we can determine the corresponding angle of rotation in degrees.

A measure of π4 radians is equivalent to a rotation of 45^(∘). This is a special angle with a sine value of exactly 1sqrt(2)

As done in Part A, we will first convert the radians to degrees in order to determine which special angle corresponds to the given measurement.

As we can see, 4π3 radians corresponds to a rotation of 240^(∘). The reference angle of this rotation is 60^(∘).

The angle 60^(∘) is a special angle with a sine value of sqrt(3)2. However, notice that 240^(∘) is in the third quadrant where the y-values are negative. That means the value of sin 4π3 is the opposite number of sqrt(3)2.

As we can see, sin 4π3 =- sqrt(3)2.

In the diagram, we see an angle that measures $6$ radians.

a unit circle with an angle that measures 6 radians

Approximate how many degrees

6

radians is without the use of a calculator. Give the answer in degrees rounded to the nearest whole number.

Use the diagram to estimate the value of

sin 6 .

Round the answer to one decimal place.

A full turn in the unit circle equals a circumference of 2π which approximates to 6.27 units. Therefore, 6 radians must be equal to an angle that is slightly less than a full turn.

To approximate 6 radians, let's view it together with an angle of 330^(∘).

The angle of 6 radians looks like it is halfway between 330^(∘) and 360^(∘). This means we can approximate how many degrees 6 radians corresponds to by adding these angles and dividing the sum by 2. 6 rad ≈ 330^(∘)+360^(∘)/2 = 345^(∘) Answers within an interval of 340^(∘) and 350^(∘) are acceptable

To estimate sin 6, recall that sine is given by the unit circle's vertical axis. Therefore, to find the sine value we must draw a horizontal line through the point on the unit circle that spans 6 radians of the circumference.

From the diagram, we see that 6 radians corresponds to a sine value of about - 0.3.

For which angle is the number of radians and degrees the same?

We want to investigate if there is any angle which has the same value in degrees and radians. If we assume such an angle has a measure in degrees of v, then the corresponding number of radians should be the following. v = v( π180) rad Let's solve this equation for v.

When v=0^(∘), the value is the same for radians and degrees.

7 5^{\circ}

arc in circle

A

and a

3 0^{\circ}

arc in circle

B

have the same length. What is the ratio of the radius

r_{A}

to the radius

r_{B} ?

Let's recall the formula for the length of an arc. s=2π r(θ/360^(∘)) We know that circle A has a sector with a central angle of 75^(∘) while the sector of circle B has a central angle of 30^(∘). Since the lengths of the arcs in the two circles are the same, we can write the following equation. 2π r_A(75^(∘)/360^(∘))=2π r_B(30^(∘)/360^(∘)) Let's simplify the equation. Our goal is to have the ratio of r_A to r_B on one side of the equation.

The ratio of the radius of circle A to the radius of circle B is 25.

Radian Measure and Arc Length

Catch-Up and Review

Investigating the Orbit of a Satellite

Radian

Relationship Between Radians and Degrees

Proof

Practice Converting Degrees Into Radians

Practice Converting Radians Into Degrees

Investigating the Radian Measure for Different Circles

Comparing Radian Measures of Concentric Circles

Using Radians to Calculate Arc Lengths

Arc Length Between the Long and Short hand of a Clock

Hint

Solution

Finding the Length of an Arc

Hint

Solution

Finding Angle Measures Given an Arc Length

Hint

Solution

Finding the Orbit of a Satellite

Hint

Solution

Radian Measure and Arc Length

Recommended exercises

	12 Theory slides
	12 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions