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Trigonometry

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.  

Exercise 1

Before we consider the values of the trig functions in the other quadrants in general we look at some examples.

Examples

Recall from the Trig Ratios page that

and .

and .

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

Hence

and

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

Hence

and

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

Hence

and

Exercise 2

Find the values of the trig functions at the angles given below. Write your answers in fractional form.
Write -1/2 for -0.5, 1/sqrt(2) for 0.707 (3d.p.). Type sqrt(a) for .

Recall that

and

Signs of Trig Functions

	We have seen that all the values of the trig functions in the first quadrant are positive. sin θ > 0 cos θ > 0 tan θ > 0
	Second quadrant: sin θ > 0 cos θ < 0 tan θ < 0
	Third quadrant: sin θ < 0 cos θ < 0 tan θ > 0
	Fourth quadrant: sin θ < 0 cos θ > 0 tan θ < 0

Summary:

To remember which trig functions are positive:

All Students Take Calculus

Exercise 3

<< Measurement of Angles (Degrees and Radians)  | Trigonometry Index | Calculators and Trigonometric Functions >>