Chapter 12 Application of Trigonometry

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Notations About Triangles The opposite sides of angles are labelled by small letters, e.g. a, b and c. The angles at vertices are denoted by capital letters, e.g. A, B and C.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Area of Triangle h  c sin B In  ABC, Area of  ABC h  b sin C

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry In  ABC, Area of  ABC Formula for the Area of Triangle

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry The following relation among sides and angles are called the sine formula. Sine Formula

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Ambiguous Cases When two sides and one non-included angle are given, is it possible to construct a unique triangle?

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Case I  No triangle can be constructed.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Case II  Only a right-angled triangle can be constructed.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Case III  Two triangles can be constructed.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Case IV  Only one triangle can be constructed.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Cosine Formula

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Cosine Formula The cosine formula can be used to solve triangles if the following conditions are given:  Two sides and their included angle  Three sides

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Heron’s Formula  Heron (around ), a Greek mathematician, focused on the studies of mechanics and engineering.  He found that the area of a triangle can be calculated by only using its three sides.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Heron’s Formula

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Angles of Elevation and Depression     

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Bearing In compass bearing, all bearings are measured from the north or from the south in an acute angle to the east or to the west. e.g.The compass bearing of B from A is N35  E. The compass bearing of A from B is S35  W. In true bearing, all bearings are measured from the north in a clockwise direction. A three-digit integer is used to represent the integral part of true bearing. e.g.The true bearing of B from A is 035 . The true bearing of A from B is 215 .

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Angle between a Line and a Plane A normal of a plane is a line perpendicular to every line lying on the plane. e.g. BC is a normal of . The projection of a point on a plane is the foot of perpendicular drawn from the point to the plane. e.g. The projection of B on  is C. The angle between a line and a plane is the angle between the given line and its projection on the given plane. e.g. The angle between AB and  is .

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Line of Greatest Slope A line of greatest slope on an inclined plane is a line perpendicular to the line of intersection between the inclined plane and the horizontal plane. e.g. AB is a line of greatest slope on  ’.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry Angle between Two Planes The angle between two planes is the angle between any two lines, one on each plane, perpendicular to the common edge of the two planes. e.g. The angle between  and  ’ is .

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 12 Application of Trigonometry The bearing of point P, above the horizontal plane ABCD, from O is the bearing of its projection P’ on ABCD from O. e.g. The compass bearing of P from O is N  W. Bearing of Points Not Lying on a Horizontal Plane

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