Transcript Trigonometry (2) Contents 11.1 Area of Triangles 11.2 Sine Formula 11.3 Cosine Formula Home 11.4 Applications in Two-dimensional Problems 11 Trigonometry (2) 11.1 Area of Triangles A.

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Trigonometry (2)
Contents
11.1 Area of Triangles
11.2 Sine Formula
11.3 Cosine Formula
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11.4 Applications in Two-dimensional
Problems
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Trigonometry (2)
11.1 Area of Triangles
A. Area Formula of Triangles
In Fig. 11.6, we take BC as the base and AD as the height of the triangle.
1
Area of ABC   base  height
2
1
 ah..................(*)
2
h
 sin C 
b
 h  b sin C
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Substituting h = b sin C into (*), we have
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1
Area of ABC  ab sin C
2
Fig. 11.6
If C is a right angle, the area
1
of ABC becomes ab.
2
In this case, b is the height
of the triangle.
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Trigonometry (2)
11.1 Area of Triangles
B. Heron’s Formula
Another important formula for calculating the area of a triangle is Heron’s
formula.
Heron’s Formula
Area of triangle  s ( s  a)(s  b)(s  c) where s 
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1
(a  b  c).
2
For any triangles with the length of all the three sides known, Heron’s
formula can be used to calculate its area.
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Trigonometry (2)
11.2 Sine Formula
The Sine Formula states that:
For any triangle, the length of a side is directly
proportional to the sine of its opposite angle.
Or mathematically, the sine formula can be expresses as:
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sin A sin B sin C


a
b
c
or
a
b
c


sin A sin B sin C
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Trigonometry (2)
11.2 Sine Formula
A. Solving a Triangle with Two Angles and Any Side Given
1.
If any two angles (A and B) of a triangle and a
side (a) opposite to one of the angles are given,
we can use the sine formula directly to find b:
a
b

sin A sin B
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2.
Fig. 11.35
If any two angles (A and B) of a triangle are given, but the given side c
is not an opposite side, we should find the third angle (C) first, then we
can use the sine formula:
b
c

sin B sin C
or
a
c

sin A sin C
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Trigonometry (2)
11.2 Sine Formula
B. Solving a Triangle with Two Sides and One Non-included Angle Given
Example 11.5T
In ABC, a = 16 cm, b = 14 cm and B = 48.
(a) Find the possible values of A.
(b) How many triangles can be formed?
Solution:
(a) By sine formula,
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(b) Two triangles can be formed.
16
14

sin A sin 48
16 sin 48
 0.8493
14
A  58.1 or 180  58.1
sin A 
 58.1 or 121.9 (correct to 1 decimal place)
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Trigonometry (2)
11.3 Cosine Formula
The following formulas are known as the cosine formulas:
Cosine Formulas
a 2  b 2  c 2  2bc cos A
b 2  a 2  c 2  2ac cos B
c 2  a 2  b 2  2ab cos C
Notes:
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When A  90, a 2  b 2  c 2  2bc cos 90
 b 2  c 2 ( cos 90  0).
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So Pythagoras’ Theorem is a special case of cosine formula for right-angled
triangles.
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Trigonometry (2)
11.4 Applications in Two-dimensional Problems
A. Angle of Elevation and Angle of Depression
When we observe an object above us, the angle  between our line of sight
and the horizontal is called the angle of elevation (see Fig. 11.76(a)).
Fig. 11.37(a)
When we observe an object below us, the angle  between the line of sight
and the horizontal is called the angle of depression (see Fig. 11.76(b)).
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Fig. 11.37(b)
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Trigonometry (2)
11.4 Applications in Two-dimensional Problems
B. Bearing
When using compass bearing, all angles are measured from north (N) or
South (S), thus the bearing is represented in the form
N E, N W, Sθ E or Sθ W where 0    90.
(a) The compass bearing of A from O is N30E.
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(b) The compass bearing of B from O is S40W.
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Fig. 11.79(a)
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Trigonometry (2)
11.4 Applications in Two-dimensional Problems
B. Bearing
When using true bearing, all angles are measured from the north in a clockwise
direction. The bearing is expressed in the form , where 0   < 360.
For example, in Fig. 11.79(b), O, C and D lie on the same plane.
(a) The bearing of C from O is 050.
(b) The bearing of D from O is 210.
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Fig. 11.79(b)
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