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Trigonometric Functions as Projections

The sine function

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?

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 \(\theta\).

Exercises

  1. What is the range of the sine function? (That is, what are the minimum and maximum values \(\sin(\theta)\) can take for any \(\theta\)?)
  2. 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.)

Check your answers

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