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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?