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 are defined from a right-angled triangle, a triangle where one internal angle is 90 ∘∘.
Consider a right-angled triangle.
In the right-angled 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 labelled the hypotenus, the side opposite θθ is labelled opposite, the side next to θθ is labelled adjacent. Note that the choice of non-90 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.
We define the trigonometric functions, also known as trigonometric identities, as:
sin
θ
=
o
p
p
o
s
i
t
e
h
y
p
o
t
e
n
u
s
e
cos
θ
=
a
d
j
a
c
e
n
t
h
y
p
o
t
e
n
u
s
e
tan
θ
=
o
p
p
o
s
i
t
e
a
d
j
a
c
e
n
t
sin
θ
=
o
p
p
o
s
i
t
e
h
y
p
o
t
e
n
u
s
e
cos
θ
=
a
d
j
a
c
e
n
t
h
y
p
o
t
e
n
u
s
e
tan
θ
=
o
p
p
o
s
i
t
e
a
d
j
a
c
e
n
t
(5)These functions relate the lengths of the sides of a right-angled triangle to its interior 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 right-angled triangles! Always check to make sure your triangle has a right-angle before you use them, otherwise you will get the wrong answer. We will find ways of using our knowledge of right-angled triangles to deal with the trigonometry of non right-angled 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' theorum. The square of the hypothenuse (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.
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
(10) - Step 2. Rearrange the question to solve for xx :
⇒
x
=
100
×
sin
50
∘
⇒
x
=
100
×
sin
50
∘
(11) - Step 3. Use your calculator to find the answer :
Use the sin
button on your calculator
⇒
x
=
76
.
6
m
⇒
x
=
76
.
6
m
(12)
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
(13) - Step 2. Calculate the fraction as a decimal number :
⇒
tan
θ
=
0
.
5
⇒
tan
θ
=
0
.
5
(14) - Step 3. Use your calculator to find the angle :
Since you are finding the angle,
use tan-1tan-1
on your calculator
Don't forget to set your calculator to `deg' mode!
⇒
θ
=
26
.
6
∘
⇒
θ
=
26
.
6
∘
(15)
The following videos provide a summary of what you have learnt so far.
