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4.1 – Radian and Degree Measure

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4.1 – Radian and Degree Measure 4.1 { Radian and Degree Measure 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 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. 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.
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