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You probably already know one definition of the sine of an angle as being OPPOSITE/HYPOTENUSE in a right triangle. This definition is fine as far as it goes, but it has limits. One problem with this definition is that the largest angle in a right triangle, other than the right angle, must be less than 90°. Grab your calculator. Ask it for the sine of 100°. Does it give an error?

If the OPPOSITE/HYPOTENUSE definition was all there was to the sine function, asking your calculator for sin(100°) should have generated an error. Because it didn't, your calculator must be using a different definition of the sine function. What is this definition?

Refer to the interactive below.

Let *P* be a point defined by the intersection of a ray
drawn at an angle \(\theta\) anti-clockwise from the positive
*x-*axis with the unit circle.

Project a line parallel to the
*x-*axis and passing through *P*.

The sine of angle
\(\theta\) is defined as the value of
*y* where this line passes through the *y-*axis (or more simply, the y-coordinate of the point).

Drag the red dot to explore the value of \(\sin(\theta)\) for different values of \(\theta\).

- What is the
*range*of the sine function? (That is, what are the minimum and maximum values \(\sin(\theta)\) can take for any \(\theta\)?) - The
*x-*and*y-*axes divide the plane into four quadrants. In which of these quadrants is \(\sin(\theta)\) positive, and in which is \(\sin(\theta)\) negative? Sketch a diagram with the*x-*and*y-*axes shown, and put a '+' or '-' in each quadrant. (As always, make sure you clearly label your sketch.)

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