Transcript File

5. Applications of trigonometry
Cambridge University Press
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 G K Powers 2013
Right-angled trigonometry
The mnemonic ‘SOH CAH TOA’ is used to determine the
trigonometric ratio.
SOH: Sine-Opposite-Hypotenuse
CAH: Cosine-Adjacent-Hypotenuse
TOA: Tangent-Opposite-Adjacent
hypotenuse
opposite

adjacent
HSC Hint – A trigonometry problem with a right-angled
triangle usually involves SOH CAH TOA.
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Compass bearing
A direction given by stating the angle either side of north
or south. For example, a compass bearing of N52ºE is
found by measuring an angle of 52º from the north
direction towards the east side.
HSC Hint – The angle in a compass bearing is always
measured with the vertical line (north-south).
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True bearing
A direction given by measuring the angle clockwise from
north. For example, 120ºT is the direction measured 120º
clockwise from north. It is the same bearing as S60ºE.
HSC Hint – The word ‘from’ in bearing questions refers to
the starting point or central point in a diagram.
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Trigonometry with obtuse angles
 Acute
angle (0 to 90º)
sin θ ‒ positive
cos θ ‒ positive
tan θ ‒ positive
 Obtuse
angle (90 to 180º)
sin θ ‒ positive
cos θ ‒ negative
tan θ ‒ negative
HSC Hint – The context of the question may indicate
whether the angle is acute or obtuse.
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The sine rule
Sine rule is used in a non-right angled triangle given
information about two sides and two angles.
 To find a side use
a
b
c
=
=
sin A sin B sinC
 To find an angle use
sin A sin B sinC
=
=
a
b
c
HSC Hint – The sine rule requires a side and the sine of a
matching angle. Matching sides are always
opposite each other on a diagram.
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Area of a triangle
Area of a triangle is half the product of two sides
multiplied by the sine of the angle between the two sides
(included angle)
A =
1 bcsin A
2
A =
1 acsin B
2
A =
1 absinC
2
side
angle
side
HSC Hint – The formula for the area of a triangle requires
sides a and b to form angle C. Sides a and b are
the arms of the angle.
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The cosine rule
Cosine rule is used in a non-right angled triangle given
information about three sides and one angle.
 To find a side use
a 2 = b2 + c2 - 2bccos A
 To find an angle use
b2 + c 2 - a 2
cos A =
2bc
HSC Hint – Don’t forget to take the square root of the value
for the expression b2 + c2 - 2bccos A
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Miscellaneous problems
1.
Read the question and underline key terms.
2.
Draw a diagram and label the information.
3.
If a right triangle, use SOH CAH TOA.
4.
If the triangle does not have a right-angle:
 use the sine rule if given two sides and three angles
 use the cosine rule if given three sides and one angle.
5.
Check that the answer is reasonable and units are correct.
HSC Hint – Problems involving two triangles require the result
of one calculation to be used in another triangle
that share a common side or angle.
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Offset survey
An offset (or traverse) survey measures distances along a
suitable diagonal or traverse. The perpendicular distances
from the traverse to the vertices of the shape are called the
offsets.
HSC Hint –Perimeter in an offset survey is calculated
using Pythagoras’ theorem.
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Radial survey
Radial survey involves measuring the angles and sides
taken from a central point. There are two methods:
 Plane-table radial survey
Angle between each
radial line is measured.
 Compass radial survey
True bearing of each
corner is measured with
a compass.
HSC Hint – Perimeter in a radial survey is calculated
using the cosine rule.
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