Trigonometric Functions for General Angles
Suppose the coordinate of a point P on the unit circle is ( x, y).
And the angle enclosed by OP and the positive x-axis is θ.
When P is in the first quadrant, θ is an acute angle.
By using the definitions of trig ratios, we have
sin θ = y, cos θ = x
provided x ≠ 0
We will take this as a general definition of trig functions for any angle θ.
We first consider P on the x and y- axes.
P = (x,y) = ( 1, 0) sin 0 = 0, cos 0 = 1, .
P = (x,y) = ( 0, 1) , is undefined.
Before we consider the values of the trig functions in the other quadrants in general we look at some examples.
Recall from the Trig Ratios page that
Let us find the values of these trig function at θ = 90º + 30º = 120º. This is in the second quadrant, where x < 0 and y > 0. From the following picture we see that
We find the values of the trig function at θ = 180º + 30º = 210º. This is in the third quadrant, where x < 0 and y < 0. From the following picture we see that
We find the values of the trig function at θ = 270º + 30º = 300º = -60º. This is in the fourth quadrant, where x > 0 and y < 0. From the following picture we see that
Find the values of the trig functions at the angles given below. Write
your answers in fractional form.
Signs of Trig Functions
To remember which trig functions are positive:
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