Reference Triangles on Standard Plane

Move the slider to change the Standard angle and see

a triangle with sides labeled for trigonometric functions

O = Opposite Side
A = Adjacent Side
H = Hypotenuse

the Reference angle in each corresponding quadrant,
the coordinates of a point at the tip of the triangle with H=r=1.

r = radius = distance from origin, (0,0)

A play button at the bottom will show the angle increasing. The tip of the triangle moves around a circle of radius 1. The right side shows the value of Sine, Cosine and Tangent as the angle is changed. A trace of the values is also shown so you can see how the functions vary. ctrl-F will clear the traces.

This is one of the main the principles behind using the trigonometric functions and inverse trigonometric functions for any angle. The Adjacent side is the absolute value of the x coordinate and the Opposite side is the absolute value of y coordinate of the right triangle. The definitions of the trigonometric functions on a plane are:

In this illustration

, so the trigonometric functions can be calculated directly from the coordinates of the triangle tip. Adjustments can then be made for the sign of the trigonometric functions based on which quadrant the reference triangle is in. Also, shown are the values of trigonometric functions for both reference and standard angles. In the results list, negative values for functions of the standard angle,

, are highlighted in red. The graph on the right shows the values of the trigonometric functions as a function of the standard angle. A point is added for each degree if the function is checked. How do the signs of the trigonometric functions vary in the different quadrants? How could the mnemonic "CAST" be used to remember which functions are positive starting in quadrant IV? How would you find the inverse function of tangent in quadrant III ?

Reference Triangles on Standard Plane

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