We are familiar with a function of the form f(x)f(x) where ff is the function and xx is the argument. Examples are:
f
(
x
)
=
2
x
(exponential function)
g
(
x
)
=
x
+
2
(linear function)
h
(
x
)
=
2
x
2
(parabolic function)
f
(
x
)
=
2
x
(exponential function)
g
(
x
)
=
x
+
2
(linear function)
h
(
x
)
=
2
x
2
(parabolic function)
(4)The basis of trigonometry are the trigonometric functions. There are three basic trigonometric functions:
These are abbreviated to:
These functions can be defined from a rightangled triangle, a triangle where one internal angle is 90 ∘∘.
Consider a rightangled triangle.
In the rightangled triangle, we refer to the lengths of the three sides according to how they are placed in relation to the angle θθ. The side opposite to the right angle is labeled the hypotenuse, the side opposite θθ is labeled opposite, the side next to θθ is labeled adjacent. Note that the choice of non90 degree internal angle is arbitrary. You can choose either internal angle and then define the adjacent and opposite sides accordingly. However, the hypotenuse remains the same regardless of which internal angle you are referring to (because it is ALWAYS opposite the right angle and ALWAYS the longest side).
We define the trigonometric functions, also known as trigonometric identities, as:
sin
θ
=
opposite
hypotenuse
cos
θ
=
adjacent
hypotenuse
tan
θ
=
opposite
adjacent
sin
θ
=
opposite
hypotenuse
cos
θ
=
adjacent
hypotenuse
tan
θ
=
opposite
adjacent
(5)These functions relate the lengths of the sides of a rightangled triangle to its interior angles.
The trig ratios are independent of the lengths of the sides of a triangle and depend only on the angles, this is why we can consider them to be functions of the angles.
One way of remembering the definitions is to use the following mnemonic that is perhaps easier to remember:
Table 2
Silly Old Hens 
S
in
=
O
pposite
H
ypotenuse
S
in
=
O
pposite
H
ypotenuse

Cackle And Howl 
C
os
=
A
djacent
H
ypotenuse
C
os
=
A
djacent
H
ypotenuse

Till Old Age 
T
an
=
O
pposite
A
djacent
T
an
=
O
pposite
A
djacent

You may also hear people saying Soh Cah Toa. This is just another way to remember the trig functions.
The definitions of opposite, adjacent and hypotenuse are only applicable when you are working with rightangled triangles! Always check to make sure your triangle has a rightangle before you use them, otherwise you will get the wrong answer. We will find ways of using our knowledge of rightangled triangles to deal with the trigonometry of non rightangled triangles in Grade 11.
 In each of the following triangles, state whether aa, bb and cc are the hypotenuse, opposite or adjacent sides of the triangle with respect to the marked angle.
 Complete each of the following, the first has been done for you
a)sinA^= opposite hypotenuse =CBACb)cosA^=c)tanA^=a)sinA^= opposite hypotenuse =CBACb)cosA^=c)tanA^=
(6)d)sinC^=e)cosC^=f)tanC^=d)sinC^=e)cosC^=f)tanC^=
(7)  Complete each of the following without a calculator:
sin60=cos30=tan60=sin60=cos30=tan60=
(8)sin45=cos45=tan45=sin45=cos45=tan45=
(9)
For most angles θθ, it is very difficult to calculate the values of sinθsinθ, cosθcosθ and tanθtanθ. One usually needs to use a calculator to do so. However, we saw in the above Activity that we could work these values out for some special angles. Some of these angles are listed in the table below, along with the values of the trigonometric functions at these angles. Remember that the lengths of the sides of a right angled triangle must obey Pythagoras' theorem. The square of the hypotenuse (side opposite the 90 degree angle) equals the sum of the squares of the two other sides.
Table 3

0
∘
0
∘

30
∘
30
∘

45
∘
45
∘

60
∘
60
∘

90
∘
90
∘

180
∘
180
∘

cos
θ
cos
θ

1 
3
2
3
2

1
2
1
2

1
2
1
2

0 

1

1

sin
θ
sin
θ

0 
1
2
1
2

1
2
1
2

3
2
3
2

1 
0 
tan
θ
tan
θ

0 
1
3
1
3

1 
3
3




0 
These values are useful when asked to solve a problem involving trig functions without using a calculator.
Each of the trigonometric functions has a reciprocal that has a special name. The three reciprocals are cosecant (or cosec), secant (or sec) and cotangent (or cot). These reciprocals are given below:
cosecθ=
1
sinθ
secθ=
1
cosθ
cotθ=
1
tanθ
cosecθ=
1
sinθ
secθ=
1
cosθ
cotθ=
1
tanθ
(10)We can also define these reciprocals for any right angled triangle:
cosec
θ
=
hypotenuse
opposite
sec
θ
=
hypotenuse
adjacent
cot
θ
=
adjacent
opposite
cosec
θ
=
hypotenuse
opposite
sec
θ
=
hypotenuse
adjacent
cot
θ
=
adjacent
opposite
(11)
Find the length of x in the following triangle.
 Step 1. Identify the trig identity that you need :
In this case you have an angle (50∘50∘), the opposite side and the hypotenuse.
So you should use sinsin
sin
50
∘
=
x
100
sin
50
∘
=
x
100
(12)
 Step 2. Rearrange the question to solve for xx :
⇒
x
=
100
×
sin
50
∘
⇒
x
=
100
×
sin
50
∘
(13)
 Step 3. Use your calculator to find the answer :
Use the sin
button on your calculator
⇒
x
=
76
.
6
m
⇒
x
=
76
.
6
m
(14)
Find the value of θθ in the following triangle.
 Step 1. Identify the trig identity that you need :
In this case you have the opposite side and the hypotenuse to the angle θθ.
So you should use tantan
tan
θ
=
50
100
tan
θ
=
50
100
(15)
 Step 2. Calculate the fraction as a decimal number :
tan
θ
=
0
.
5
tan
θ
=
0
.
5
(16)
 Step 3. Use your calculator to find the angle :
Since you are finding the angle,
use tan1tan1
on your calculator
Don't forget to set your calculator to `deg' mode!
θ
=
26
.
6
∘
θ
=
26
.
6
∘
(17)
In the previous example we used tan1tan1. This is simply the inverse of the tan function. Sin and cos also have inverses. All this means is that we want to find the angle that makes the expression true and so we must move the tan (or sin or cos) to the other side of the equals sign and leave the angle where it is. Sometimes the reciprocal trigonometric functions are also referred to as the 'inverse trigonometric functions'. You should note, however that tan1tan1 and cotcot are definitely NOT the same thing.
The following videos provide a summary of what you have learnt so far.