 Home > College of Sciences > Institute of Fundamental Sciences > Maths First > Online Maths Help > Trigonometry > Trig Functions for General Angles SEARCH MASSEY  # 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

Without using a calculator compute the following (type ND if it is undefined):

 Angle θ sin θ cos θ tan θ π -π / 2

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 Angle θ sin θ cos θ tan θ 135° 150° 240° -45° ## 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

Type + if the sign of the trig function of an angle is positive, or type −for negative.
Type ND if the trig function is undefined.
Type 0 when neither negative, positive or undefined.

 Angle θ sin θ cos θ tan θ 1 2π/3 270° -180° -50° -2π/3