Lecture 1: Basics of trigonometry (surveying)

1
BASICS OF
TRIGONOMETRY
Lecture 1

WHAT IS ANGLE
 An angle is defined as the amount of turn
between two straight lines that share a common
end point.
 Angles are measured in degrees.
 The symbol used for degrees is a little circle °
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ANGLE CLASSIFICATION
 Angles measured in surveying are classifies as
either Horizontal or Vertical, depending on the
plane in which they are observed.
 The instrument used in the measurement of
angles (Theodolite, Total Station)
3

TERMS ASSOCIATED WITH ANGLES
VERTEX - The vertex of an angle is the common point
where the two lines meet.
ARM - The arms of an angle or sides are the lines that
make up the angle.
DEGREES - The size of the angle is measured in degrees
and usually denoted with the ° symbol. For example, an
angle may measure 45°.
PROTRACTOR - A tool that is used to measure angles.
4

NAMING ANGLE
To name an angle, we name any point on one ray,
then the vertex, and then any point on the other ray
We may also name this angle only by the single letter of the
vertex, for example <B.
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If the angle is not exactly to the next degree it can be expressed
as a decimal (most common in math) or in degrees, minutes and
seconds (common in surveying and some navigation).
1 degree = 60 minutes 1 minute = 60 seconds
 = 25°48'30"
degrees
minutes
seconds
Let's convert the
seconds to
minutes
30"
"
60
'
1
 = 0.5'
7
Angle measurements
 = 25°48'30"
48.5'
'
60
1
 = .808°
= 25°48.5'
= 25.808°


initial side
radius of circle is r
r
r
arc length is
also r
r
This angle measures
1 radian
Given a circle of radius r with the vertex of an angle as the
center of the circle, if the arc length formed by intercepting the
circle with the sides of the angle is the same length as the
radius r, the angle measures one radian.
ANOTHER WAY TO MEASURE ANGLES IS USING
WHAT IS CALLED RADIANS.
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arc length radius measure of angle
important: angle measure
must be in radians to use
formula!
Find the arc length if we have a circle with a radius of 3
meters and central angle of 0.52 radian.
3
 = 0.52
arc length to find is in black
s = r
3 0.52 = 1.56 m
What if we have the measure of the angle in degrees? We
can't use the formula until we convert to radians, but how?
s = r
ARC LENGTH S OF A CIRCLE IS FOUND WITH THE
FOLLOWING FORMULA:
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conversion from degrees to radians
Let's start with
the arc length
formula
s = r
If we look at one revolution
around the circle, the arc
length would be the
circumference. Recall that
circumference of a circle is
2r
2r = r
cancel the r's
This tells us that the
radian measure all the
way around is 2. All the
way around in degrees is
360°.
2 = 
2  radians = 360°
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𝑅𝑎𝑑𝑖𝑎𝑛 = 𝐷𝑒𝑔𝑟𝑒𝑒 𝑥
𝜋 radians
180°
To convert degree to radian:

It is customary to use small letters in the Greek alphabet
to symbolize angle measurement.
 




alpha beta gamma
theta
phi delta
GREEK SIGNS
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In Plane Trigonometry there are 6 trigonometric functions, best
explained with reference to a right triangle.
Angles are denoted by the symbol < and sides a, b and c are
known. The < C is a right angle:
The initial 3 basic trig functions are defined as:
sine of < A (written as sin A)
cosine of < A (written as cos A)
tangent of < A (written as tan A)
TRIGONOMETRY FUNCTIONS
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Pythagoras developed the proof of the geometric theorem that
states that in a right angled triangle the square of the hypotenuse
is equal to the sum of the squares of the other two sides:
a² +b² = c²
Depending on which values are known the equation can be re-
written to solve for either a, b or c:
PYTHAGORAS’ THEOREM
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➢ As we have discussed only right triangles which are solved
using the basic functions. However, there are triangles that do
not have a right angle – these are called scalene triangles.
➢ Scalene triangles can have all acute interior angles (<90°) or can
have one angle that is obtuse (>90°).
➢ Problems as applied to scalene triangles normally fall into the
following categories:
2 known sides, 1 known angle
2 known angles, 1 known side
3 known sides with no known angle
SCALENE TRIANGLES
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Depending on the data provided, the solution of scalene triangles
can be achieved by one of the following methods:
1. Constructed Right Triangles
2. Sine Law
3. Cosine Law
SCALENE TRIANGLES
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1-Constructed Triangles
Example:
In triangle ABC <B = 34°18’30”, b = 26.860 m and c = 42.225 m.
Find side a, <A and <C.
Solutions:
A perpendicular is dropped from A to meet side BC at D. Two right
angled triangles are formed ABD and ACD.
First consider triangle ABD
SCALENE TRIANGLES
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Now consider triangle ACD:
SCALENE TRIANGLES
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Calculate the distance between the two building corners Q and R
and the perpendicular distance PS to the building using the interior
angle measured at P and the two measured distances PQ and PR.
HOMEWORK
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2-The Sine Law states that in any triangle ABC with angles A, B and C,
and corresponding sides a, b and c:
This law is useful when computing the remaining sides of a triangle if two
angles and a side are known. It can also be used when two sides and one
of the non-enclosed angles are known.
3-The Cosine Law states that in any triangle ABC with angles A, B and
C, and corresponding sides a, b and c, the following equation is true:
This law is useful in computing the angle when all three sides are known:
SCALENE TRIANGLES
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Sine & Cosine Law Examples
A surveyor is set up at point A on the playing field and measures the
angle between two building corners B and C together with the
distances AB and AC.
Calculate the distance between the two building corners and the two
interior angles at points B and C.
SCALENE TRIANGLES
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Solution:
Step 1. Use the Cosine Law to find the distance BC (side a)
Step 2. Use the Sine Law to solve <B
Step 3. <C may then be deduced i.e.
<C = 180° – 66° 28’ 45” – 78° 24’ 29” = 35° 06’ 46”
Alternatively <C may also be calculated by the Sine Law
SCALENE TRIANGLES
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Surveyors will be required to solve triangle applications frequently,
having measured angles and distances, examples include the
following:
Right Triangle Applications
➢ Determination of elevation using vertical angles and slope
distances (Trigonometric Levelling).
➢ Coordinate Calculations – ΔE, ΔN values
Scalene Triangle Applications
➢ Calculation of distances around or through immoveable objects.
➢ Determination of the height or depth of an object relative to a known elevation.
➢ Determination of distance to an inaccessible location.
TRIANGLE APPLICATIONS IN SURVEYING
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Lecture 1: Basics of trigonometry (surveying)

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Lecture 1: Basics of trigonometry (surveying)