Cambridge International AS and A Level Mathematics Pure Mathematics 1

Trigonometry

P1^

7

The tangent graph

The value of tan θ can be worked out from the definition tan θ =

y

x or by using

tan θ =

sin

cos

θ.

θ

You have already seen that tan θ is undefined for θ = 90°. This is also the case for

all other values of θ for which cos θ = 0, namely 270°, 450°, ..., and −90°, −270°, ...

The graph of tan θ is shown in figure 7.16. The dotted lines θ = ±90° and

θ = 270° are asymptotes. They are not actually part of the curve. The branches of

the curve get closer and closer to them without ever quite reaching them.

Note

The graph of tan θ is periodic, like those for sin θ and cos θ, but in this case the

period is 180°. Again, the curve for 0  θ  90° can be used to generate the rest of

the curve using rotations and translations.

ACTIvITy 7.2 Draw the graphs of y = sin θ, y = cos θ, and y = tan θ for values of θ between −90°

and 450°.

These graphs are very important. Keep them handy because they will be useful

for solving trigonometrical equations.

Note

Some people use this diagram to help them remember

when sin, cos and tan are positive, and when they are

negative. A means all positive in this quadrant, S means sin

positive, cos and tan negative, etc.

–90° 90° 180° 270° 360° θ

y

0°

Figure 7.16

These are asymptotes.

Figure 7.17

A

T C

S