Flying into Trig on a Paper Plate

Transcription

1 Flying int Trig n a Paper Plate Warm-up: 1. Label the quadrants:. Classify the fllwing angles as btuse, acute r right: a) b) 91 c) 90 d) 18. Add the fllwing fractins (withut a calculatr!) a) + 5 b) 1 8 Flying Int Trig n a Paper Plate Intr: 1. Lcate and mark the center f the circle. (Fld the paper plate in half and then in half again). We will call the center pint O. Mark yur center pint with the letter O. The unit circle is a circle with a radius equal t ne and the center at the rigin f a rectangular crdinate system.. What is the circumference f the abve circle?. Darken (mark) the flds and label the x and y axis with a clred marker. D nt use the black marker.. Lcate the pint where the psitive x-axis intersects the unit circle and label it A using the black marker. This represents the anchr pint (the beginning) 5. Using the prvided pipe cleaners, trace arund the perimeter f the circle. Meanwhile keep a tally f the number f the number f pipe cleaners that were used. (ne shuld nt be cmpletely used, mark the pint and measure it using a ruler).. Repeat step 5 three mre times, nce the furth attempt is cmplete, average the number f pipe cleaners that it tk t cmplete the circumference f the plate. 7. Divide yur average by the length f ne pipe cleaner. Cmpare yur answer t.8. Hw clse were yu?

2 8. If the radius f the plates dubled, apprximately hw many pipe cleaners with length equivalent t the radius f thse plates wuld yu expect t need t circle the circumference? Definitin f Radian Measure: Angles in Trignmetry are measured in and. 180 = radians Fill in the fllwing table, using yur paper plate: Arc Measurement Angle, in Radians Angle, in Degrees 1. Using the abve table, hw wuld yu cnvert an angle frm radians t degrees?. Hw wuld yu cnvert frm degrees t radians? Example Cnvert frm degrees t radians: a) 00 b) c) 90 d) 18 Example Cnvert frm radians t degrees: a). 1 7 b). c). 17. radian 5

3 Angles in Standard Psitin Examples: Draw the fllwing angles in standard psitin: a) θ =!! b) α =!!! c) β =!!! d) γ =!!! Cterminal Angles are angles in standard psitin (angles with the initial side n the psitive x- axis) that have a cmmn terminal side. Fr example 0, 0 and 90 are all cterminal. Examples: Name a cterminal angle t each f the fllwing angles. a) θ =!! b) α =!!! c) β =!!! d) γ =!!!

4 Reference Angles: The reference angle is the acute angle frmed by the terminal side f the given angle and the x-axis. Reference angles may appear in all fur quadrants. Angles in quadrant I are their wn reference angles. Remember: The reference angle is measured frm the terminal side f the riginal angle "t" the x-axis (nt the y-axis). Examples: Name the reference angle: a) θ =!! b) α =!!! c) β =!!! d) γ =!!!

5 5 Practice State the quadrant in which the terminal side f the angle lies Find a c-terminal angle that is between zer and 0 r If each angle has the given measure and is in standard psitin, determine the quadrant in which each terminal side lies. Name the c-terminal angle Find ne psitive and ne negative c-terminal angle Find the reference angle fr each given angle

6 Warm-UP: Using the unit circle: Flying int Trig n a Paper Plate: Creating the Unit Circle 1. What is the length f OA? What are the crdinates f the pint where:. the psitive x-axis intercepts the UNIT CIRCLE?. the psitive y-axis intercepts the UNIT CIRCLE?. the negative x-axis intercepts the UNIT CIRCLE? 5. the negative y-axis intercepts the UNIT CIRCLE? Label these crdinates n yur unit circle. Activity One ALWAYS check with the instructr BEFORE putting anything n yur unit circle. 1. Remembering the special right triangles frm Gemetry, we can find the x and y crdinate f each crrespnding value n the family triangle.. In the issceles right triangle, remember that the length f the sides have the rati 1:1: (Nte: rder is x: y: hyptenuse). Hwever, since we are using the UNIT CIRCLE, ur hyptenuse is 1. S we will divide each f the three values by t be 1. in rder t maintain the prper rati and get the hyptenuse. The new rati values are : :. 5. We als must remember t ratinalize the denminatrs, s the new values are : :.. Using the rati f an Issceles Triangle, cmplete the fllwing triangle. Label the arc measures, the sides f the triangle, and the crdinates f the arc measures. 7. Label the crdinates n yur unit circle.

7 7 Activity Tw 1. The rati f the lengths f the sides in a 0 : 0 : 90 triangle is :1: (Nte: rder is x: y: hyptenuse). Hwever, since we are using the UNIT CIRCLE, ur hyptenuse is 1. S we will divide each f the three values by in rder t maintain the prper rati and get the hyptenuse t be 1.. The new rati values are : :.. Cmplete the fllwing triangle. Label the arc measures, the sides f the triangle, and the crdinates f the arc measures. 5. Label the crdinates n yur unit circle. Activity Fur ALWAYS check with the instructr BEFORE putting anything n yur unit circle. 1. The rati f the lengths f the sides in a 0 : 0 : 90 triangle is 1: : (Nte: rder is x: y: hyptenuse). Hwever, since we are using the UNIT CIRCLE, ur hyptenuse is 1. S we will divide each f the three values by in rder t maintain the prper rati and get the hyptenuse t be 1.. The new rati values are : :.. Cmplete the fllwing triangle. Label the arc measures, the sides f the triangle, and the crdinates f the arc measures. 5. Label the crdinates n yur unit circle.

8 Hmewrk - Use the fllwing diagram t cmplete the unit circle. Include radian measures, degree measures and crdinates. 8

9 9 Warm-up: Simplify: / Remember fr every pint n the unit circle, we can draw a right triangle: Using Right Triangle Trig, write dwn the fllwing in terms f x and y: sinθ = csθ = tanθ = Use yur paper plate t find the exact values f the fllwing trig functins: 1. cs 5. sin. cs. sin 5 5. sin0. cs90 7. sin ( 70 ) 8. cs 7 9. tan 10. tan tan 1. tan0

10 10 Hmewrk Use yur Unit Circle t find the fllwing: 1. sin 5. cs 5. sin. cs 5. cs0. sin ( 90 ) 7. cs0 8. sin 9. tan 10. tan tan 1. tan G nline t research answers t the fllwing questins: 1. S far we have used three trig functins sine, csine and tangent. But actually there are six trig functins! What are they?. What are the reciprcal identities? List them here:

11 11 Warm-Up: Frm Last ntes. Remember fr every pint n the unit circle, we can draw a right triangle: Using Right Triangle Trig, write dwn the fllwing in terms f x and y: sinθ = csθ = tanθ = cscθ = secθ = ctθ = Find each exact value using the Unit Circle. 1. Sin. Cs 5. 5 Tan. Cs Sin 00. Tan 0 7. Sin 8. Cs Sin 10. Cs Tan 1. Csc 0 1. Sec 9 1. Ct Ct 5 1. Csc Sec ( 70 ) 18. Tan ( 10 ) 19. Cs 0 0. Sin 1

12 Test Review If each angle has the given measure and is in standard psitin, determine the quadrant in which each terminal side lies Change each degree measure t radian measure Change each radian measure t degree measure Find ne psitive and ne negative angle that are c-terminal with the given angle Find the reference angle fr each angle with the given measure

13 1 Refer t the unit circle t help yu evaluate the fllwing trignmetric functins. 7. Sec Cs Ctan ( 5 ) 0. Csc 1. sin. Tan. Sec 11. Tan ( 150 ) 5. Cs. Tan ( 00 ) 7. Ctan 7 8. sin ( 0 ) 9. Sec 0. Cs 5