Unit 6 Introduction to Trigonometry Degrees and Radians (Unit 6.2) - PowerPoint PPT Presentation

Unit 6 – Introduction to Trigonometry Degrees and Radians (Unit 6.2) William (Bill) Finch Mathematics Department Denton High School

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Lesson Goals When you have completed this lesson you will: ◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds. W. Finch DHS Math Dept Radian/Degree 2 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Lesson Goals When you have completed this lesson you will: ◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds. W. Finch DHS Math Dept Radian/Degree 2 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Lesson Goals When you have completed this lesson you will: ◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds. W. Finch DHS Math Dept Radian/Degree 2 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Lesson Goals When you have completed this lesson you will: ◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds. W. Finch DHS Math Dept Radian/Degree 2 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Lesson Goals When you have completed this lesson you will: ◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds. W. Finch DHS Math Dept Radian/Degree 2 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Lesson Goals When you have completed this lesson you will: ◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds. W. Finch DHS Math Dept Radian/Degree 2 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Angles in Standard Position An angle in standard position : y Terminal ◮ starts on positive x -axis Positive (initial side) x ◮ rotates counter-clockwise for Initial positive angles ◮ rotates clockwise for negative y angles Terminal ◮ often named with Greek letters Initial ◮ theta . . . θ x ◮ alpha . . . α Negative ◮ beta . . . β W. Finch DHS Math Dept Radian/Degree 3 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Degree Measure y 90 ◦ 120 ◦ 60 ◦ 135 ◦ 45 ◦ 150 ◦ 30 ◦ x (0 ◦ ) 180 ◦ 360 ◦ 0 ◦ 210 ◦ 330 ◦ 225 ◦ 315 ◦ 240 ◦ 300 ◦ 270 ◦ W. Finch DHS Math Dept Radian/Degree 4 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Degree-Minutes-Seconds (DMS) A fraction of a degree can be expressed as a decimal fraction, but historically the degree was divided into minutes ( ′ ) and seconds ( ′′ ) . 1 ◦ = 60 ′ 1 ′ = 60 ′′ and For example, 32 . 125 ◦ = 32 ◦ 7 ′ 30 ′′ Read “ 32 degrees, 7 minutes, and 30 seconds.” W. Finch DHS Math Dept Radian/Degree 5 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Degree-Minutes-Seconds (DMS) A fraction of a degree can be expressed as a decimal fraction, but historically the degree was divided into minutes ( ′ ) and seconds ( ′′ ) . 1 ◦ = 60 ′ 1 ′ = 60 ′′ and For example, 32 . 125 ◦ = 32 ◦ 7 ′ 30 ′′ Read “ 32 degrees, 7 minutes, and 30 seconds.” W. Finch DHS Math Dept Radian/Degree 5 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Example 1 Convert to decimal degrees. a) 25 ◦ 15 ′ b) 12 ◦ 10 ′ 33 ′′ W. Finch DHS Math Dept Radian/Degree 6 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Calculator Instructions – TI-84 W. Finch DHS Math Dept Radian/Degree 7 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Example 2 Convert to degree-minutes-seconds. a) 48 . 4 ◦ b) 21 . 456 ◦ W. Finch DHS Math Dept Radian/Degree 8 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Calculator Instructions – TI-84 W. Finch DHS Math Dept Radian/Degree 9 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Radian Measure One radian is the measure of a y central angle θ that intercepts an arc s equal in length to the radius r of the circle: r s θ θ = s x r r where θ is measured in radians. Note that in the diagram above the radius r of the circle is the same length as the arc s intercepted by the two radii, so θ = 1 rad when s = r . W. Finch DHS Math Dept Radian/Degree 10 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Radian Measure The circumference of a circle is y one revolution around the circle. C = 2 π r θ s = 2 π r x s r = 2 π θ = 2 π θ ≈ 6 . 28 A central angle θ that is one revolution is 2 π radians. W. Finch DHS Math Dept Radian/Degree 11 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Radian Measure The circumference of a circle is y one revolution around the circle. C = 2 π r θ s = 2 π r x s r = 2 π θ = 2 π θ ≈ 6 . 28 A central angle θ that is one revolution is 2 π radians. W. Finch DHS Math Dept Radian/Degree 11 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Radian Measure The circumference of a circle is y one revolution around the circle. C = 2 π r θ s = 2 π r x s r = 2 π θ = 2 π θ ≈ 6 . 28 A central angle θ that is one revolution is 2 π radians. W. Finch DHS Math Dept Radian/Degree 11 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Radian Measure The circumference of a circle is y one revolution around the circle. C = 2 π r θ s = 2 π r x s r = 2 π θ = 2 π θ ≈ 6 . 28 A central angle θ that is one revolution is 2 π radians. W. Finch DHS Math Dept Radian/Degree 11 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Radian Measure The circumference of a circle is y one revolution around the circle. C = 2 π r θ s = 2 π r x s r = 2 π θ = 2 π θ ≈ 6 . 28 A central angle θ that is one revolution is 2 π radians. W. Finch DHS Math Dept Radian/Degree 11 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Radian Measure The circumference of a circle is y one revolution around the circle. C = 2 π r θ s = 2 π r x s r = 2 π θ = 2 π θ ≈ 6 . 28 A central angle θ that is one revolution is 2 π radians. W. Finch DHS Math Dept Radian/Degree 11 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Radian Measure One revolution around a circle is slightly more than 6 radians. y 2 rad 1 rad s = r r 3 rad x 6 rad 4 rad 5 rad W. Finch DHS Math Dept Radian/Degree 12 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Radian Measure y 90 ◦ 120 ◦ 60 ◦ 135 ◦ 45 ◦ π 2 2 π π 150 ◦ 3 30 ◦ 3 3 π π 4 4 5 π π 6 6 π x 180 ◦ 2 π 360 ◦ 0 ◦ 7 π 11 π 6 5 π 6 7 π 4 4 210 ◦ 330 ◦ 4 π 5 π 3 3 3 π 2 225 ◦ 315 ◦ 240 ◦ 300 ◦ 270 ◦ W. Finch DHS Math Dept Radian/Degree 13 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Special Angles – Learn Them! y y π 2 π x x y y 2 π x x 3 π 2 W. Finch DHS Math Dept Radian/Degree 14 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Special Angles – Learn Them! y y 3 π π 4 4 x x y y x x 5 π 7 π 4 4 W. Finch DHS Math Dept Radian/Degree 15 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Special Angles – Learn Them! y y 2 π π 3 3 x x y y x x 4 π 5 π 3 3 W. Finch DHS Math Dept Radian/Degree 16 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary Special Angles – Learn Them! y y 5 π π 6 6 x x y y x x 7 π 11 π 6 6 W. Finch DHS Math Dept Radian/Degree 17 / 35