Sine and Cosine
This problem is a follow on to the problems Round and Round and Round and Where Is the Dot?
Use your calculator to find decimal values for the following:
$\sin 50 ^{\circ}$, $\cos 40 ^{\circ}$
$\sin 70 ^{\circ}$, $\cos 20 ^{\circ}$
$\sin 15^{\circ}$, $\cos 75^{\circ}$
What do you notice and why does that happen?
Look at the film below - does that fit with your description?
Look especially at the last part of the film, you may also find the Pause button useful.
The film suggests a way to understand Sine and Cosine ratios (or lengths, if the hypotenuse has length one), for angles beyond the $0 ^{\circ}$ to $90^{\circ}$ range, in other words beyond angles which occur in right-angled triangles.
Which of these statements do you think are true?:
$\sin 150 ^{\circ}= \sin 30^{\circ}$ (notice that 180 - 30 = 150)
$\sin 150 ^{\circ}= \sin 330 ^{\circ}$
$\sin 150 ^{\circ}= \sin 210^{\circ}$
$\sin 30^{\circ}= \sin 330 ^{\circ}$
$\cos 30 ^{\circ}= \cos 330 ^{\circ}$
$\cos 50^{\circ}= \cos 130 ^{\circ}$
$\sin 150 ^{\circ}= \cos 30 ^{\circ}$
$\sin 150 ^{\circ}= \cos 60 ^{\circ}$
$\sin 300 ^{\circ}= \cos 30 ^{\circ}$
You could use your calculator to check.
What other relationships can you find?
Can you make some general statements?
The Flash "movie" allows students to see the flow of function values as the dot circles the centre point or origin. Students can be encouraged to see the "spot" values provided by a calculator as snap-shots of this flow.
Check that students grasp that an angle of zero is when the dot is at (1,0), that the red line corresponds to the Sine value and the blue line to the Cosine value.
There may be a need to guide students into noticing that there is a the change of sign, and noticing when it occurs.