Everything Maths Grade 11

CHAPTER 17. TRIGONOMETRY 17.4

y

x

1

− 1

360 − 180 − 180 360

Step 2 :

Notice that this line touches the graph four times. This means that thereare four

solutions to the equation.

Step 3 :

Read off the x values of those intercepts from the graph as x =− 330 ◦;− 210 ◦;

30 ◦and 150 ◦.

y

x

1

− 1

360 270 − 180 − − 90 − 90 180 270 36 0

This method can be time consuming and inexact. We shall now look athow to solve these problems
algebraically.

Solution using CAST diagrams EMBDH

The Sign of the Trigonometric Function

The first step to findingthe trigonometry of anyangle is to determine the sign of the ratio for a given
angle. We shall do thisfor the sine function first and then do the samefor the cosine and tangent.

In Figure 17.11 we havesplit the sine graph intofour quadrants, each 90 ◦wide. We call them quad-
rants because they correspond to the four quadrants of the unit circle. Wenotice from Figure 17.11that
the sine graph is positive in the 1 stand 2 ndquadrants and negativein the 3 rdand 4 th. Figure 17.12
shows similar graphs forcosine and tangent.

All of this can be summed up in two ways. Table17.7 shows which trigonometric functions are positive
and which are negativein each quadrant.

A more convenient wayof writing this is to notethat all functions are positive in the 1 stquadrant,
only sine is positive inthe 2 nd, only tangent in the 3 rdand only cosine in the 4 th. We express this