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Maths Notes
Trigonometry
2. Sin, Cos, Tan
2. Sin, Cos and Tan
1. The Crucial Point about Sin, Cos and Tan
Just like Pythagoras Theorem, all the work we will be doing with Sin, Cos and Tan only works
with RIGHT-ANGLED TRIANGLES
So… if you don’t have a right-angled triangle, you might just have to add a line or two to make
one!
2. Checking your Calculator is in the Correct Mode
Every now and again calculators have a tendency to do stupid things, one of which is slipping into
the wrong mode for sin, cos and tan questions, giving you a load of dodgy answers even though
you might be doing everything perfectly correctly!
Here is the check: Work out: sin 30
And if you get an answer of 0.5, you are good to go!
If not, you will need to change into degrees (DEG) mode.
Each calculator is different, but here’s how to do this on
mine:
3. Labelling the Sides of a Right-Angled Triangle
• Before you start frantically pressing buttons on your calculator, you must work out which one
of the trig ratios (sin, cos or tan) than you need, and to do this you must be able to label the
sides of your right-angled triangle correctly.
• This is the order to do it:
1. Hypotenuse (H) – the longest side, opposite the right-angle
2. Opposite (O) – the side directly opposite the angle you have been given / asked to work out
3. Adjacent (A) – the only side left!
O

H
H
A
O
H

A
A
O


Note:
is just the Greek letter Theta, and it is used for unknown angles, just
like x is often used for unknown lengths!
4. The Two Ways of Solving Trigonometry Problems
Both methods start off the same:
1. Label your right-angled triangle
2. Tick which information (lengths of sides, sizes of angles) you have been given
3. Tick which information you have been asked to work out
4. Decide whether the question needs sin, cos or tan
The difference comes now, where you actually have to go on and get the answer.
Both of the following methods are perfectly fine, just choose the one that suits you best!
(a) Use the Formulas and Re-arrange
If you are comfortable and confident re-arranging formulas, then this method is for you!
Just learn the following formulas:
Sine  
Sin  
Opposite
Hypotenuse
O
H
Cosine  
Cos  
Adjacent
Hypotenuse
A
H
Tangent  
Tan  
Opposite
Adjacent
O
A
Now just substitute in the two values you do know, and re-arrange the equation to find the
value you don’t know!
(b) Use the Formula Triangles
This is a clever little way of solving any trig problem.
Just make sure you can draw the following triangles from memory:
o
sin θ
a
h
cos θ
o
h
tan θ
a
A good way to remember these is to use the initials, reading from left to right:
S O H
C A H
T O A
And make up a way of remembering them (a pneumonic, is the posh word!). Now, I know a good
one about a horse, but it might be a bit too rude for this website…
Anyway, once you have decided whether you need sin, cos or tan, just put your thumb over the
thing (angle or side) you are trying to work out, and the triangle will magically tell you exactly
what you need to do!
Finding Opposite:
Finding Hypotenuse:
o = Sin θ x h
h = a ÷ Cos θ
Examples
1.
?
√
500
12 cm
Okay, here we go:
1. Label the sides
2. Tick which information we have been given…
which I reckon is the angle and the Adjacent
3. Tick which information we need… which I
reckon is the Opposite side
4. Decide whether we need sin, cos or tan … well,
looking above, the only one that contains both O
and A is… Tan!
5. Now we place our thumb over the
thing we need to work out, which is the
Opposite:
√
O
?
H
√
500
√ A 12 cm
6. And now we know how to do it!
o = Tan θ x a
14.3 cm (1dp)
2.
3.1 m
260
√
?
O
Okay, here we go:
1. Label the sides
2. Tick which information we have been given…
which I reckon is the angle and the Adjacent
3.1 m
√ A
√
260
?
H
√
3. Tick which information we need… which I
reckon is the Hypotenuse side
4. Decide whether we need sin, cos or tan … well,
looking above, the only one that contains both A
and H is… Cos!
5. Now we place our thumb over the
thing we need to work out, which is the
Hypotenuse:
6. And now we know how to do it!
h = a ÷ Cos θ
3.45 m (2dp)
3.
8 mm
?
√
6 mm
8.1 mm
H√
√
Okay, here we go:
?
A
1. Label the sides
2. Tick which information we have been given…
which I reckon is the Hypotenuse and the Opposite
3. Tick which information we need… which I reckon
is the angle
4. Decide whether we need sin, cos or tan … well,
looking above, the only one that contains both O and
H is… Sin!
5. Now we place our thumb over the
thing we need to work out, which is the
angle… or Sin θ
6 mm
O
√
6. Okay, be careful here:
Sin θ = o ÷ h
Sin θ = 0.740740740740…
But that’s not the answer! We don’t want to know what Sin
θ is, we want to know what θ is, so we must use “inverse
sin” to leave us with just θ on the left hand side:
47.790 (2dp)
4.
10 cm
Now, we have a problem here… we don’t have a
right-angled triangle! But we can easily make one
appear from this isosceles triangle by adding a
vertical line down the centre, and then we can
carry on as normal…
400
H
O
10 cm
?
1. Label the sides
2. Tick which information we have been given…
which I reckon is the angle and the Opposite
3. Tick which information we need… which I reckon
is the Adjacent
4. Decide whether we need sin, cos or tan … well,
looking back, the only one that contains both O and
A is… Tan!
5. Now we place our thumb over the
thing we need to work out, which is the
Adjacent
700
A
?
6. And now we know how to do it!
a = o ÷ Tan θ
3.639702…
But that’s not the answer! We’ve only worked out half of
the base of the isosceles triangle! So we need to double
this to give us our true answer of:
7.28 cm (2dp)
Good luck with
your revision!