Chapter 11 Trigonometry (2)

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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

11 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 Home Content

Substituting h = b sin C into (*), we have 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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11 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 = Home Content

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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11 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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11 Trigonometry (2) 11.2 Sine Formula A. Solving a Triangle with Two Angles and Any Side Given •

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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•

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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11 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:

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(a) By sine formula, (b) Two triangles can be formed. 16 14 = sin A sin 48° 16 sin 48° sin A = ≈ 0.8493 14 A = 58.1° or 180° − 58.1° = 58.1° or 121.9° (correct to 1 decimal place)

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11 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

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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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11 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 θ 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 φ between the line of sight and the horizontal is called the angle of depression (see Fig. 11.76(b)). Home Content

Fig. 11.37(b)

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11 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 Nθ E, Nθ W, Sθ E or Sθ W where 0 < θ < 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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11 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 θ, where 0° ≤ θ < 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°. Home Content

Fig. 11.79(b)

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