Trigonometrical Ratios
The
ratio (fraction expressed as a decimal)
of the length of one side of a
RIGHT ANGLED TRIANGLE to the length of
another side.
This gives a unique value for every angle.
Sides are identified by reference to the
angle under consideration.
Opposite ( to angle A)
Identify the sides
A
Adjacent (Next to Angle A)
Adjacent (Next to Angle B)
Identify the sides -2
B
Opposite ( to angle B)
The Ratios


SINE (A) =
OPP
HYP
COSINE (A) =
ADJ
HYP
A

TANGENT (A) =
OPP
ADJ
=
SIN(A)
COS(A)

Rearrange to find length of a side:

Opposite
OPP
HYP
Hence:
=
OPP
Opposite
Using the Ratios
Sin (A)
A
Adjacent
=
Sin (A) X HYP
Using the Ratios -2
=
COS(A)
Hence:
ADJ
= Cos (A) X HYP
Opposite
ADJ
HYP
A
Adjacent
Remember




Remember if you know length of HYPOTENEUSE
and an ANGLE
and ADJACENT is involved consider COSINE
formula
if OPPOSITE is involved consider SINE formula
Only OPPOSITE and ADJACENT and ANGLE ? then consider TANGENT formula
Angles





Angles will be usually expressed in
Sexagesimal system (Degrees, Minutes and
seconds)
Most usual cause of mistakes
Get used to using the D M S or
 ‘ ‘’
buttons on your calculator.
Ensure Mode is on Degrees!
If no conversion button – just a few more buttons
to be pressed!
Angles -2
Consider conversion from decimal to Sexagesimal:
 E.g. 35.5678123
 ( Note that we must use at least 6 decimal places for
“seconds” accuracy)
 = 35 plus a fraction of a degree
i.e. 0.5678123 which can be converted to minutes by
MULTIPLYING by 60
 =0.5678123 x 60 = 34.068738 Minutes
 This is 34 Minutes plus a fraction of a minute
i.e. 0.068738 which can be converted to seconds by
MULTIPLYING by 60
 =0.068738 x 60 = 4.12428 = 4 seconds
 So 35.5678123 = 35 34’ 4’’

Angles -3



Consider the conversion of Sexagesimal to
decimals
E.g. 35 34’ 4’’
Integer part = 35

Fractional part (minutes): 34’ / 60
=0.5666667
Fractional part (seconds): 4’’/(60x60) =0.0011111
Add the two fractional part of a degree: =0.5677778

Hence 35 34’ 4’’ = 35.5677778


Examples




Plan length from measured slope length and angle of
inclination:
S = 25.567
S
Angle A = 11 35’ 40’’
Find Plan length D
A
D







Adj, Hyp and angle – hence use COSINE
Cos(A) = Adj/Hyp
Adj = Hyp x Cos(A)
Hence D = s x Cos (A)
D = 25.567 x Cos(11 35’ 40’’)
D = 25.567 x 0.979594
D = 25.045