Sketching AnglesSketching Angles

And Coterminal AnglesAnd Coterminal Angles

Standard PositionStandard Position

An angle is in standard position if its An angle is in standard position if its vertex is at the origin and its initial vertex is at the origin and its initial side is along the positive x-axis.side is along the positive x-axis.

Link to Link to

http://www.mathopenref.com/degreehttp://www.mathopenref.com/degrees.htmls.html

Quadrantal Angles Quadrantal Angles

Quadrant I

0° < θ < 90°

Quadrant II

90° < θ < 180°

Quadrant III

180° < θ < 270°

Quadrant IV

270° < θ < 360°

90°

180°

270°

360°

Angles in standard position having their terminal sides along the x-axis or y-axis are called Quadrantal angles. (shown in red)

Let’s PracticeLet’s Practice

Take your white board and sketch Take your white board and sketch each angle in standard position. each angle in standard position. Indicate its rotation by a curved Indicate its rotation by a curved arrow. Classify each angle by it’s arrow. Classify each angle by it’s quadrant.quadrant.

1) 50°1) 50° 2) 120 °2) 120 ° 3) 260 °3) 260 ° 4) 300 °4) 300 °

AnswersAnswers

Take your white board and sketch Take your white board and sketch each angle in standard position. each angle in standard position. Indicate its rotation by a curved Indicate its rotation by a curved arrow. Classify each angle by it’s arrow. Classify each angle by it’s quadrant.quadrant.

1) 50° I1) 50° I 2) 120 ° II2) 120 ° II 3) 260 ° III3) 260 ° III 4) 300 ° IV4) 300 ° IV

ReviewReview

A A quadrantal anglequadrantal angle is one that is one that has its terminal side on one of has its terminal side on one of the coordinate axes.the coordinate axes.

What are the four quadrantal What are the four quadrantal angles?angles?

4 Quadrantal angles4 Quadrantal angles

90° 90°

180° (1/2 revolution)180° (1/2 revolution)

270° 270°

360° (1 complete revolution)360° (1 complete revolution)

Practice - Sketch each angle in Practice - Sketch each angle in standard position. Indicate it’s rotation standard position. Indicate it’s rotation by a curved arrow. Classify each angle by a curved arrow. Classify each angle

by its quadrant. If the angle is a by its quadrant. If the angle is a quadrantal angle, say so. This is on quadrantal angle, say so. This is on

your worksheet.your worksheet.

1) 135°1) 135° 2) -2402) -240 3) 403) 40 4) -3004) -300 5) -2705) -270 6) 3156) 315 7) 2907) 290 8) -908) -90 10) 49510) 495 11) -81011) -810 12) 75012) 750

ANSWERSANSWERS

1) II1) II 2) II2) II 3) I3) I 4) I4) I 5) Quadrantal Angle5) Quadrantal Angle 6) IV6) IV 7) IV7) IV 8) Quadrantal 8) Quadrantal

AngleAngle 10) II10) II 11) Quadrantal Angle 12) I11) Quadrantal Angle 12) I

Coterminal AnglesCoterminal Angles

Coterminal anglesCoterminal angles are angles are angles that have the same terminal side that have the same terminal side (unlimited). (unlimited).

60° and 420°

(360 + 60)

One revolution + 60

HAVE THE SAME TERMINAL SIDE

The UNIT CIRCLEThe UNIT CIRCLE

This unit circle will be essential to This unit circle will be essential to our study in trig.our study in trig.

You will want to study it daily You will want to study it daily because we will be adding new because we will be adding new pieces of information to it pieces of information to it constantly. Yes, you will need to constantly. Yes, you will need to memorize it!memorize it!

Today we will label the positive Today we will label the positive angles.angles.

Try this exampleTry this example

Find the angle of smallest positive Find the angle of smallest positive measure coterminal with this angle.measure coterminal with this angle.

908°908°

Try this exampleTry this example

Find the angle of smallest positive Find the angle of smallest positive measure coterminal with this angle.measure coterminal with this angle.

908°908°

Hint: subtract off 360 as many times Hint: subtract off 360 as many times as need to obtain an angle with as need to obtain an angle with measure greater than 0 but less than measure greater than 0 but less than 360360

Try this exampleTry this example

Find the angle of smallest positive Find the angle of smallest positive measure coterminal with this angle.measure coterminal with this angle.

908°908° 908 – 360 = 548 908 – 360 = 548

548 – 360 = 188548 – 360 = 188

So 188° is coterminal with an So 188° is coterminal with an angle of 908°angle of 908°

How about a negative How about a negative angle?angle?

Find the angle of smallest positive Find the angle of smallest positive measure coterminal with this angle.measure coterminal with this angle.

-75°-75°

-75°

How about a negative How about a negative angle?angle?

Find the angle of smallest positive Find the angle of smallest positive measure coterminal with this angle. measure coterminal with this angle. HINT: use of rotation of 360°HINT: use of rotation of 360°

-75°-75°

360 + (-75) = 285°360 + (-75) = 285°

-75°285°

CoterminalCoterminal

Let n represent any integer, then all angles Let n represent any integer, then all angles coterminal with an angle of 150° can be coterminal with an angle of 150° can be expressed as 150° + (n x 360°)expressed as 150° + (n x 360°)

N = 0 then 150° + (0 x 360°) = 150 °N = 0 then 150° + (0 x 360°) = 150 ° N = 1 then 150° + (1 x 360°) = 510 °N = 1 then 150° + (1 x 360°) = 510 ° N = 2 then 150° + (2 x 360°) = 870 °N = 2 then 150° + (2 x 360°) = 870 ° N = -1 then 150° + (-1 x 360°) = -210 °N = -1 then 150° + (-1 x 360°) = -210 °

Find the angle of smallest Find the angle of smallest positive measure positive measure

coterminal with each anglecoterminal with each angle If positive angle, keep subtracting If positive angle, keep subtracting

off 360° until it is in the range 0 < off 360° until it is in the range 0 < θθ < 360° < 360°

If negative angle, then add 360°If negative angle, then add 360°

1)1) -40 °-40 ° 5) 539 °5) 539 °

2)2) -98 °-98 ° 6) 699 °6) 699 °

3)3) -125 °-125 ° 7) 850 °7) 850 °

4)4) -203 °-203 ° 8) 1000 °8) 1000 °

AnswersAnswers

1) 320 °1) 320 ° 5) 179 °5) 179 ° 2) 262 °2) 262 ° 6) 339 °6) 339 ° 3) 235 °3) 235 ° 7) 130 °7) 130 ° 4) 157 °4) 157 ° 8) 280 °8) 280 °

Now try in radiansNow try in radians

If positive angle, keep subtracting If positive angle, keep subtracting off 2off 2ππ until it is in the range 0 < until it is in the range 0 < θθ < 2< 2ππ

If negative angle, then add 2If negative angle, then add 2ππ

1)1) 1212ππ/5/5

2)2) 1717ππ/4/4

3)3) -7-7ππ/3/3

AnswersAnswers

1) 121) 12ππ/5 - 10/5 - 10ππ/5 = /5 = 2 2ππ /5 /5 2) 172) 17ππ/4 - 8/4 - 8ππ/4 = 9/4 = 9ππ/4 now too large/4 now too large So subtract off 2 So subtract off 2 ππ and get 9 and get 9ππ/4 -8/4 -8ππ/4 = /4 = ππ/4 /4 3) -73) -7ππ/3 + /3 + 66ππ/3 = - /3 = - ππ /3 Now less than 0 so /3 Now less than 0 so

add 2 add 2 ππ - - ππ/3 + 6 /3 + 6 ππ /3 = 5 /3 = 5ππ/3/3

Fractions are fun!Fractions are fun!

HomeworkHomework

Page 453 #2-36 evenPage 453 #2-36 even

Packet p. 6Packet p. 6 Practice unit circlePractice unit circle