4.1 – Radian and Degree Measure

Home , Degree (angle), Radian

Accelerated Pre-Calculus

Mr. Niedert

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 1 / 27 2 Radian Measure

3 Degree Measure

4 Applications

4.1 – Radian and Degree Measure

1 Angles

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 2 / 27 3 Degree Measure

4 Applications

4.1 – Radian and Degree Measure

1 Angles

2 Radian Measure

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 2 / 27 4 Applications

4.1 – Radian and Degree Measure

1 Angles

2 Radian Measure

3 Degree Measure

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 2 / 27 4.1 – Radian and Degree Measure

1 Angles

2 Radian Measure

3 Degree Measure

4 Applications

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 2 / 27 The starting position of the ray is the initial side of the angle. The position after the rotation is the terminal side of the angle. The vertex of the angle is the endpoint of the ray.

Angles

An angle is determine by rotating a ray about it endpoint.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 3 / 27 The position after the rotation is the terminal side of the angle. The vertex of the angle is the endpoint of the ray.

Angles

An angle is determine by rotating a ray about it endpoint. The starting position of the ray is the initial side of the angle.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 3 / 27 The vertex of the angle is the endpoint of the ray.

Angles

An angle is determine by rotating a ray about it endpoint. The starting position of the ray is the initial side of the angle. The position after the rotation is the terminal side of the angle.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 3 / 27 Angles

An angle is determine by rotating a ray about it endpoint. The starting position of the ray is the initial side of the angle. The position after the rotation is the terminal side of the angle. The vertex of the angle is the endpoint of the ray.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 3 / 27 Angles in Standard Position

An angle is said to be in standard position if it situated on a coordinate plane with the vertex located at the origin and the initial side coinciding with the positive x-axis.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 4 / 27 Angles in Standard Position

An angle is said to be in standard position if it situated on a coordinate plane with the vertex located at the origin and the initial side coinciding with the positive x-axis.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 4 / 27 Negative angles are generated by clockwise rotation.

Positive and Negative Angles

Positive angles are generated by counterclockwise rotation.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 5 / 27 Positive and Negative Angles

Positive angles are generated by counterclockwise rotation. Negative angles are generated by clockwise rotation.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 5 / 27 In the figure below, notice that angles α and β have the same initial side and terminal side. This means that these angles are coterminal. Specifically, coterminal angles have different measure, but result in the same angle when in standard position.

Coterminal Angles

Angles are sometimes labeled with Greek letters such as α (alpha), β (beta), and θ (theta).

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 6 / 27 Speciﬁcally, coterminal angles have diﬀerent measure, but result in the same angle when in standard position.

Coterminal Angles

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 6 / 27 Coterminal Angles

Angles are sometimes labeled with Greek letters such as α (alpha), β (beta), and θ (theta). In the figure below, notice that angles α and β have the same initial side and terminal side. This means that these angles are coterminal. Specifically, coterminal angles have different measure, but result in the same angle when in standard position.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 6 / 27 Radian Measure

Deﬁnition of Radian One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle. Algebraically, this means that s θ = r where θ is measured in radians.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 7 / 27 This means that for each of the angles in the circle can be expressed as a portion (or fraction) of 2π.

Derivation of Radians

One revolution around a circle of radius r corresponds to an angle of 2π radians because s 2πr θ = = = 2π radians. r r

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 8 / 27 Derivation of Radians

One revolution around a circle of radius r corresponds to an angle of 2π radians because s 2πr θ = = = 2π radians. r r

This means that for each of the angles in the circle can be expressed as a portion (or fraction) of 2π.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 8 / 27 Quadrants

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 9 / 27 Sketching and Finding Coterminal Angles

Example 3π a Sketch the angle θ = and find a coterminal angle. 4 13π b Sketch the angle θ = and find a coterminal angle. 6 2π c Sketch the angle θ = − and find a coterminal angle. 3

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 10 / 27 Sketching and Finding Coterminal Angles

Practice 9π a Sketch the angle θ = and find a coterminal angle. 4 5π b Sketch the angle θ = and find a coterminal angle. 6 3π c Sketch the angle θ = − and find a coterminal angle. 4

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 11 / 27 Complementary and Supplementary Angles

Practice If possible, ﬁnd the complement and supplement of each of the angles below. π a 6 5π b 6

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 12 / 27 4.1 – Radian and Degree Measure (Part 1 of 3) Assignment

pg. 290-291 #8-24 even

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 13 / 27 ◦ 1 One degree (1 ) is equivalent to a rotation of 360 of a complete revolution about the vertex.

Degrees

Degrees are the most frequent way we measure angles and indicated by the ◦ symbol.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 14 / 27 Degrees

Degrees are the most frequent way we measure angles and indicated by the ◦ symbol. ◦ 1 One degree (1 ) is equivalent to a rotation of 360 of a complete revolution about the vertex.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 14 / 27 π Notice that 360◦ = 2π rad =⇒ 1◦ = rad. 180

Conversions Between Degrees and Radians π rad To convert degrees to radians, multiply degrees by . 180◦ 180◦ To convert radians to degrees, multiply radians by . π rad Overall, just keep in mind that since 360◦ = 2π rad then 180◦ = π rad.

From here on out, if a unit of measure for an angle is not indicated, then assume the angle is in radians. Degrees will need to be indicated using the ◦ symbol from here on out.

Converting Between Degrees and Radians

Because 2π radians corresponds to one complete revolution, degrees, and radians are related by the equation 360◦ = 2π rad.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 15 / 27 Conversions Between Degrees and Radians π rad To convert degrees to radians, multiply degrees by . 180◦ 180◦ To convert radians to degrees, multiply radians by . π rad Overall, just keep in mind that since 360◦ = 2π rad then 180◦ = π rad.

From here on out, if a unit of measure for an angle is not indicated, then assume the angle is in radians. Degrees will need to be indicated using the ◦ symbol from here on out.

Converting Between Degrees and Radians

Because 2π radians corresponds to one complete revolution, degrees, and radians are related by the equation 360◦ = 2π rad. π Notice that 360◦ = 2π rad =⇒ 1◦ = rad. 180

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 15 / 27 Overall, just keep in mind that since 360◦ = 2π rad then 180◦ = π rad.

From here on out, if a unit of measure for an angle is not indicated, then assume the angle is in radians. Degrees will need to be indicated using the ◦ symbol from here on out.

Converting Between Degrees and Radians

Because 2π radians corresponds to one complete revolution, degrees, and radians are related by the equation 360◦ = 2π rad. π Notice that 360◦ = 2π rad =⇒ 1◦ = rad. 180

Conversions Between Degrees and Radians π rad To convert degrees to radians, multiply degrees by . 180◦ 180◦ To convert radians to degrees, multiply radians by . π rad

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 15 / 27 From here on out, if a unit of measure for an angle is not indicated, then assume the angle is in radians. Degrees will need to be indicated using the ◦ symbol from here on out.

Converting Between Degrees and Radians

Because 2π radians corresponds to one complete revolution, degrees, and radians are related by the equation 360◦ = 2π rad. π Notice that 360◦ = 2π rad =⇒ 1◦ = rad. 180

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 15 / 27 Converting Between Degrees and Radians

Because 2π radians corresponds to one complete revolution, degrees, and radians are related by the equation 360◦ = 2π rad. π Notice that 360◦ = 2π rad =⇒ 1◦ = rad. 180

From here on out, if a unit of measure for an angle is not indicated, then assume the angle is in radians. Degrees will need to be indicated using the ◦ symbol from here on out.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 15 / 27 Converting from Degrees to Radians

Practice Convert each of the following from degrees to radians. Express as a multiple of π. a 135◦ b 540◦ c −270◦

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 16 / 27 Converting from Radians to Degrees

Practice Convert each of the following from radians to degrees. Round to the nearest hundredth if necessary. π a − 2 9π b 2 c2

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 17 / 27 This is read as “108 degrees, 16 minutes, and 20 seconds,” just like it is with time. We can express this angle in decimal degree form as seen below. 16 20 108◦1602000 = 108 + + ≈ 108.2722◦ 60 602

We can convert an angle already in decimal degree form, such as 345.12◦, as seen below. 345.12◦ =⇒ a little more than 345◦ .12 × 60 min = 7.2 min .2 × 60 s = 12s ◦ ◦ 0 00 ∴ 345.12 = 345 7 12

Degrees-Minutes-Seconds Sometimes you may see angles in degree form expressed as 108◦1602000.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 18 / 27 We can express this angle in decimal degree form as seen below. 16 20 108◦1602000 = 108 + + ≈ 108.2722◦ 60 602

Degrees-Minutes-Seconds Sometimes you may see angles in degree form expressed as 108◦1602000. This is read as “108 degrees, 16 minutes, and 20 seconds,” just like it is with time.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 18 / 27 We can convert an angle already in decimal degree form, such as 345.12◦, as seen below. 345.12◦ =⇒ a little more than 345◦ .12 × 60 min = 7.2 min .2 × 60 s = 12s ◦ ◦ 0 00 ∴ 345.12 = 345 7 12

Degrees-Minutes-Seconds Sometimes you may see angles in degree form expressed as 108◦1602000. This is read as “108 degrees, 16 minutes, and 20 seconds,” just like it is with time. We can express this angle in decimal degree form as seen below. 16 20 108◦1602000 = 108 + + ≈ 108.2722◦ 60 602

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 18 / 27 Degrees-Minutes-Seconds Sometimes you may see angles in degree form expressed as 108◦1602000. This is read as “108 degrees, 16 minutes, and 20 seconds,” just like it is with time. We can express this angle in decimal degree form as seen below. 16 20 108◦1602000 = 108 + + ≈ 108.2722◦ 60 602

We can convert an angle already in decimal degree form, such as 345.12◦, as seen below. 345.12◦ =⇒ a little more than 345◦ .12 × 60 min = 7.2 min .2 × 60 s = 12s ◦ ◦ 0 00 ∴ 345.12 = 345 7 12 Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 18 / 27 Degrees-Minutes-Seconds Sometimes you may see angles in degree form expressed as 108◦1602000. This is read as “108 degrees, 16 minutes, and 20 seconds,” just like it is with time. We can express this angle in decimal degree form as seen below. 16 20 108◦1602000 = 108 + + ≈ 108.2722◦ 60 602

We can convert an angle already in decimal degree form, such as 345.12◦, as seen below. 345.12◦ =⇒ a little more than 345◦ .12 × 60 min =7 .2 min .2 × 60 s = 12s ◦ ◦ 0 00 ∴ 345.12 = 345 7 12 Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 18 / 27 Degrees-Minutes-Seconds Sometimes you may see angles in degree form expressed as 108◦1602000. This is read as “108 degrees, 16 minutes, and 20 seconds,” just like it is with time. We can express this angle in decimal degree form as seen below. 16 20 108◦1602000 = 108 + + ≈ 108.2722◦ 60 602

Part 1: pg. 290-291 #8-24 even Part 2: pg. 291 #32-46 even, 47-54, 55-77 odd

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 19 / 27 Arc Length

Arc Length For a circle of radius r, a central angle θ intercepts an arc of length s given by s = rθ where θ is measured in radians. Note that if r = 1, then s = θ, and the radian measure of θ equals the arc length.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 20 / 27 Finding Arc Length

Practice A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240◦.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 21 / 27 Linear speed measures how fast a particle moves. Angular speed measures how fast the angle changes.

Linear and Angular Speeds

Linear and Angular Speeds Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed v of the particle is arc length s Linear speed v = = . time t Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, then the angular speed ω of the particle is central angle θ Angular speed ω = = . time t

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 22 / 27 Angular speed measures how fast the angle changes.

Linear and Angular Speeds

Linear speed measures how fast a particle moves.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 22 / 27 Linear and Angular Speeds

Linear speed measures how fast a particle moves. Angular speed measures how fast the angle changes.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 22 / 27 Finding Linear Speed

Practice The second hand of a clock is 10.2 centimeters long. Find the linear speed of the tip of this second hand as it passes around the clock face.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 23 / 27 Finding Angular and Linear Speeds

Practice A Ferris wheel with a 50-foot radius makes 1.5 revolutions per minute. a Find the angular speed of the Ferris wheel in radians per minute. b Find the linear speed of the Ferris wheel.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 24 / 27 Area of a Sector of a Circle For a circle of radius r, the area A of a sector of the circle with central angle θ is given by 1 A = r 2θ 2 where θ is measured in radians.

Sectors A sector of a circle is the region bounded by two radii of the circle and their intercepted arc.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 25 / 27 Sectors A sector of a circle is the region bounded by two radii of the circle and their intercepted arc.

Area of a Sector of a Circle For a circle of radius r, the area A of a sector of the circle with central angle θ is given by 1 A = r 2θ 2 where θ is measured in radians.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 25 / 27 Area of a Sector of a Circle

Practice A sprinkler on a golf course fairway is set to spray water over a distance of 70 feet and rotates through an angle of 120◦. Find the area of the fairway watered by the sprinkler.

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 26 / 27 4.1 – Radian and Degree Measure (Part 3 of 3) Assignment

Part 1: pg. 290-291 #8-24 even Part 2: pg. 291 #32-46 even, 47-54, 55-77 odd Part 3: pg. 292-293 #80-94 even, 97-104, 106-107

4.1 – Radian and Degree Measure Assignment pg. 290-293 #8-24 even, 32-46 even, 47-54, 55-77 odd, 80-94 even, 97-104, 106-107

Accelerated Pre-Calculus 4.1 – Radian and Degree Measure Mr. Niedert 27 / 27