Trigonometry: Deriving the Sine Function

1. Trigonometry:Deriving the Sine Function Suganya Chandrakumar & HumairaMasehoor

2. Connection to the Curriculum Course MCF3M: Functions and Applications Strand Trigonometry Expectation 2.4 Sketch the graph of f(x) = sinxfor angle measures expressed in degrees, and determine and describe its key properties (i.e., cycle, domain, range, intercepts, amplitude, period, maximum and minimum values, increasing/decreasing intervals)

3. Learning Goals Students will: • Develop a clear understanding of the unit circle • Make a connection between the unit circle and the sine function

4. Agenda for the Day Ferris Wheel Video Review on the Unit Circle Spaghetti Trig Ticket out the Door

5. Ferris Wheel While watching the video, I want you think about… When you ride on a Ferris wheel does your motion have anything in common with a wave?

6. Unit Circle Review • When you work with angles in all four quadrants, the trig ratio for those angles are computed in terms of the values x, y, & r • Where r is the radius of the circle that corresponds to the hypothesis of the right angle triangle for your angle • The x and y values on the unit circle are defined as: x = cos(ϴ) y = sin(ϴ) r = 1 P = (x,y) = (cos(ϴ), sin(ϴ))

7. Sine Function • Looking at the sin ratio in the four quadrants, we can take the input (the angle measure ϴ), “unwind” this to form the unit circle and put it on the horizontal axis of a standard graph in the x,y-plane. • Then we can take the output (value of sin(ϴ)) and use this value as the height of the function.