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 counterclockwise 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
 In the above figure, click 'reset' and 'hide details'.
 Drag the orange dot to make a large or negative angle
 Select a new trig function from the pull down menu and calculate that function for the new angle
 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
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