Sine Cosine: Triangle, Circle, Wave!
'What do Triangles, Circles and Waves have in Common?'
If you haven't been on a Ferris Wheel yet, you've probably seen one. Close your eyes, picture yourself 90m off the ground, level with the main axel of the Singapore Flyer, the world's largest Ferris Wheel (in 2011) and imagine how your vertical height changes as you start going higher and higher, in an anti-clockwise direction, around the wheel. Draw the graph of how your vertical height changes as time passes. Now, watch the video of a Ferris Wheel being constructed below. Can you see how the circle is made using triangles? Right angled triangles can be described using sine and cosine ratios, circles can be described using right angled triangles and waves are caused by circular (periodical) movements . . . Can you understand the link between triangles, circles, waves and the sine and cosine ratios?
Work through and complete the following activity sheet: Triangle Circle Waves
Once you have sketched the graphs of your movement around the Ferris Wheel, check your sketches using this virtual manipulative.
Experiment with the different checkboxes to view the various animations.
- Watch the video of the Ferris Wheel. Imagine yourself in a carriage level with the axel of the Ferris Wheel and about to start moving upwards. Sketch a graph of how you think your height changes with time.
- Now imagine yourself at the far left of your journey around the wheel, level with the axel. Sketch a graph of how you think your horizontal distance from the axel changes with time.
- Revise your sine and cosine ratios and see if you can understand how they can be used to define the x and y coordinates of any point on a circle of radius 1 unit.
- Use your calculator and pattern spotting abilities to quickly draw an exact graph of the sine and cosine ratios for differing angles.
- Can you explain the link between Triangles, Circles, Waves and the sine and cosine ratios?
- Challenge: Try using Pythagoras's Theorem to find the equation of a circle. Using your transformation of graphs knowledge, how do you think this equation could be modified if the centre of the circle was no longer at the origin, but at the point: (a) (0, 2) (b) (2, 0) (c) (2, 2) (d) (a,b)