I have a basic but rather conceptual question on trigonometry

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In this figure how can we find out geometrically that the sine of the angle $\theta$ = $AB /AO$ without using the equation sin (180 - $\theta$) = sin ($\theta$)

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1 Answer

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"The definition of sin of and angle is opposite/hypotenuse" - well that only works for a right-angled triangle, and is the beginning of the definition of the sine function.

In order to extend the definition draw a unit circle with centre at the origin. Measure the angle counterclockwise from the positive x-axis and take a point (x,y) on the circle. The sine of the angle (equivalent to opposite/hypotenuse in the first quadrant, with the hypotenuse made equal to 1) is y and the cosine of the angle is x. You can go round the circle more than once, so you can see that the functions are periodic.

This is why the functions are sometimes known as circular functions and underlies why they come in so surprisingly useful.

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