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A Favorite Recipe for Trigonometric Pasta! Created by Teresa Hall North Pole High School [email protected] 6/26/2017 Build the six trigonometric functions using pasta! Yes…pasta! And you will need a paper plate. Participants will receive teacher perspectives of the activities shared in the workshop. 9-12 Algebra Manipulative Targeted: Building models using simple materials Two overheads Materials 2 - 6 in paper plates (bigger size creates larger sinusoids) Linguine (flat spaghetti) Scotch tape 2 sheets 12x18 of colored paper or 8.5x11 xerox paper Protractor Ruler Unit circle model Teacher models Template for sin, cos (paper plate folded with 16 unit circle points labeled) Template for csc, cot (horizontal tangent lines at 90 and 270) Template for tan, sec (vertical tangent lines at 0 and 180) Why? Students who have learned the relationships between the sides of a 45-45-90 triangle and a 30-60-90 triangle and have labeled coordinates of the unit circle and have recited SOHCAH-TOA, may still need more concrete experiences with the 6 different trig ratios. This activity will help students make more connections from their prior knowledge of the unit circle and the ratios of two sides of a triangle to actually create the graphs of the six trigonometric functions. Procedure Sin & Cos: Fold a paper plate into 6ths and 4ths. Label each crease with the 16 common degrees found on the unit circle. Create and draw your axes. 6in paper plate has approximately an 18 in circumference. Every 1.5 inches equates to 15 degrees. You will actually snap off 20 pieces of pasta. The graph looks better to include 15, 165, 195, and 345 degrees. Snap pasta lengths using the paper plate (aka - unit circle) as your gauge. Tape the pasta to set of axes drawn the long paper. Draw points at the end of each pasta, connect the points to make a smooth curve. Created by Teresa Hall North Pole High School [email protected] 6/26/2017 Csc & Cot Make another set of axes like sine and cosine Start with the template for csc & cot (horizontal tangent lines at 90 and 270) Csc in the reciprocal of sine, so the opposite side needs to stay a constant one. The hypotenuse extended to a horizontal tangent line will be the length of the snapped pasta. Notice as x 90, the hyp to 1! Also as x 0, the hyp infinity! There’s lots of foreshadowing opportunities. Cot is the reciprocal of tangent, so again the opposite side needs to stay a constant one. Snapping pasta lengths using the adjacent side creates this graph. Notice as x 90, adj side 0 and as x0, adj side infinity. Sec & Tan Make another set of axes like sine and cosine Template for tan, sec (vertical tangent lines at 0 and 180) Sec in the reciprocal of cosine, so the adjacent side needs to remain a constant one. The hypotenuse extended to a vertical tangent line will be the length of the snapped pasta. Notice as x 90, the hyp to infinity. Also as x 0, the hypto one. Tan is the ratio of opposite to adjacent, to again the adjacent side need to remain a constant one. Snapping pasta lengths using the opposite side creates this graph. Notice as x90, adj side to infinity and as x0, the adj side to one. Once the graphs have been created, complete the following pages, which unite graphical, numerical, and analysis. Created by Teresa Hall North Pole High School [email protected] 6/26/2017 Y = sin(x) Sketch the sine graph below. Angle sin(x) in exact degrees value 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 What is the domain for sin x?____________ What is the range for sin x?______________ As x 0 degrees, sin x ______ As x 90 degrees, sin x ______ As x 180 degrees, sin x _____ As x 270 degrees, sin x _____ Created by Teresa Hall North Pole High School [email protected] 6/26/2017 Y = cos(x) Sketch the cosine graph below. Angle cos(x) in exact degrees value 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 What is the domain for cos x?______________ What is the range for cos x?_______________ As x 0 degrees, cos x ______ As x 90 degrees, cos x ______ As x 180 degrees, cos x _____ As x 270 degrees, cos x _____ Created by Teresa Hall North Pole High School [email protected] 6/26/2017 Y = tan(x) Angle in degrees 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 sin(x) exact value Sketch the tangent graph below. cos(x) exact value tan(x) exact value What is the domain for tan x? _________________ What is the range for tan x? __________________ As x 0 degrees, tan x ______ As x 90 degrees, tan x ______ As x 180 degrees, tan x _____ As x 270 degrees, tan x _____ Created by Teresa Hall North Pole High School [email protected] 6/26/2017 Y = csc(x) Sketch the cosecant graph below. Angle sin(x) csc(x) in exact exact degrees value value 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 What is the domain for csc x? _________________ What is the range for csc x? __________________ As x 0 degrees, csc x ______ As x 90 degrees, csc x ______ As x 180 degrees, csc x _____ As x 270 degrees, csc _____ Created by Teresa Hall North Pole High School [email protected] 6/26/2017 Y = sec(x) Sketch the secant graph below. Angle cos(x) sec(x) in exact exact degrees value value 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 What is the domain for sec x? _________________ What is the range for sec x? __________________ As x 0 degrees, sec x ______ As x 90 degrees, sec x ______ As x 180 degrees, sec x _____ As x 270 degrees, sec _____ Created by Teresa Hall North Pole High School [email protected] 6/26/2017 Y = cot(x) Sketch the cotangent graph below. Angle tan(x) cot(x) in exact exact degrees value value 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 What is the domain for cot x? _________________ What is the range for cot x? __________________ As x 0 degrees, cot x ______ As x 90 degrees, cot x ______ As x 180 degrees, cot x _____ As x 270 degrees, cot x _____ Created by Teresa Hall North Pole High School [email protected] 6/26/2017