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Trig functions of large and negative angles

Try this Drag the orange dot and note how the sine function varies with the angle. Drag the point around the origin multiple times in both directions.

The usual way to define the trigonometry functions is in a right triangle. For example, the sine function. is defined as the ratio of the opposite side to the hypotenuse. In a right triangle, the angle can only lie in the range 0..90°. In the figure above, click on 'reset'. Note that the angle B is 52°. The sine of 52° is the ratio of the opposite side AC to the hypotenuse AB. Therefore

Large angles

For large angles, the same idea applies. We just have to be sure to count the lengths that are to the left and below the origin as negative.

In the figure above, drag the point counter-clockwise around the origin, to say 210°. The sin of 210° is still defined as the opposite over hypotenuse, but here the opposite side has a negative length, so This idea works for all six trig functions. Experiment with the six functions from the pull down menu above.

Negative angles

In trigonometry negative angles go clockwise. The above definition applies to negative angles also. In the figure above drag the point clockwise. Note how the sine ratio still holds, and produces values similar to those for positive angles.

Unit Circle

If the hypotenuse is made to be one unit long, the trig functions all simplify.
For more on this see Unit Circle.

Things to try

  1. In the above figure, click 'reset' and 'hide details'.
  2. Drag the orange dot to make a large or negative angle
  3. Select a new trig function from the pull down menu and calculate that function for the new angle
  4. Click 'show details' to check your answer.

    (Note that the side lengths in the figure are rounded to whole numbers for clarity, so the result you get may be slightly different.)

Other trigonometry topics

Angles

Trigonometric functions

Solving trigonometry problems

Calculus