Everything Maths Grade 11

17.3 CHAPTER 17. TRIGONOMETRY

and that cos θ is defined as:

cos θ =

adjacent

hypotenuse

Therefore, we can write

tan θ =

opposite

hypotenuse

×

hypotenuse

adjacent

= sin θ×

1

cos θ

=

sin θ

cos θ

Tip

tanθ can also be de-
fined as:

tanθ=

sinθ

cosθ

A Trigonometric Identity EMBDC

One of the most usefulresults of the trigonometric functions is that theyare related to each other. We

have seen that tan θ can be written in termsof sin θ and cos θ. Similarly, we shall show that:

sin^2 θ + cos^2 θ = 1

We shall start by considering�ABC,

�

A

�

B

�

C

θ

We see that:

sin θ =

AC

BC

and

cos θ =

AB

BC

.

We also know from theTheorem of Pythagorasthat:

AB^2 + AC^2 = BC^2.

So we can write:

sin^2 θ + cos^2 θ =

�

AC

BC

� 2

+

�

AB

BC

� 2

=

AC^2

BC^2

+

AB^2

BC^2

=

AC^2 + AB^2

BC^2

=

BC^2

BC^2

(from Pythagoras)

= 1