Everything Maths Grade 10

Similarly, an alternative method is to calculate the hypotenuse using the theorem of Pythagoras and calculate
using:

sin =

opposite

hypotenuse

or cos =

adjacent

hypotenuse

NOTE:

We can also extend the definitions of the reciprocals in the same way.

Finding any angle

If we were to draw a circle centred on the origin (O) and passing through the pointP(x;y), then the length
from the origin to pointPis the radius of the circle, which we denoter. We denote the angle formed between
the lineOPand thex-axis as.



0

x

y

r

P(x;y)

We can rewrite all the trigonometric ratios in terms ofx,yandr. The general definitions for the trigonometric
ratios are:
sin =

y

r

cosec =

r

y

cos =

x

r

sec =

r

x

tan =

y

x

cot =

x

y

The CAST diagram

The Cartesian plane is divided into 4 quadrants in an anti-clockwise direction as shown in the diagram below.
Notice thatris always positive but the values ofxandychange depending on the position of the point in
the Cartesian plane. As a result, the trigonometric ratios can be positive or negative. The letters C, A, S and T
indicate which of the ratios are positive in each quadrant:

132 5.8. Defining ratios in the Cartesian plane