Trigonometry (Mathematics & Mathematics Extension 1)

1 Trigonometry ( Mathematics & Mathematics Extension 1). Copyright Enoch Lau 2003 Jan 2003 (re-pub Apr 2003). Trigonometry ( Mathematics & Mathematics Extension 1). The Trigonometric Ratios Hypotenuse Opposite . Adjacent In any right triangle, the trigonometric ratios are: opp sin = hyp adj cos = hyp SOH CAH TOA. opp tan = adj The Reciprocal Ratios 1 hyp cosec = sin = opp 1 hyp sec = cos = adj 1 adj cot = tan = opp Complementary Ratios SIN and COS are complementary ratios. TAN and COT are complementary ratios. sin = cos (90 ) tan = cot (90 ). cos = sin (90 ) cot = tan (90 ). Ratios of Special Angles (Exact Values). 1 1 1. sin 30 = 2 cos 60 = 2 cos 45 =. 2. 3 tan 60 = 3 tan 45 = 1. cos 30 = 2. 1. tan 30 = 1.

2 3 sin 45 =. 3 2. sin 60 = 2.. Page 1 of 5. Trigonometry ( Mathematics & Mathematics Extension 1). Copyright Enoch Lau 2003 Jan 2003 (re-pub Apr 2003). Bearings True bearings are bearings from North. They are always three digits, 290 T. Compass bearings are measured from the compass directions: North (N), South (S), East (E), West (W), N 70 E. Angles of Elevation and Depression The Angle of Elevation is the angle between the line of sight and the horizontal, looking up at something. The Angle of Depression is the angle between the line of sight and the horizontal, looking down at something.. Line of Sight Horizontal . Angle of depression = Angle of elevation (by alternate angles on parallel lines).

3 Sine Rule a b c sin A = sin B = sin C. sin A sin B sin C. a = b = c IMPORTANT: When dealing with the sine ratio, it is possible to have acute and obtuse solutions. Cosine Rule Finding sides: a 2 = b 2 + c 2 2bc cos A. b 2 = a 2 + c 2 2ac cos B. c 2 = a 2 + a 2 2ab cos C. Finding angles: b2 + c2 a2. cos A =. 2bc a + c2 b2. 2. cos B =. 2ac a + b2 c2. 2. cos C =. 2ab . Page 2 of 5. Trigonometry ( Mathematics & Mathematics Extension 1). Copyright Enoch Lau 2003 Jan 2003 (re-pub Apr 2003). Area of a triangle 1. A= ab sin C. 2. 1. Area = 2 % the product of the 2 sides % the sine of the included angle Angles of any magnitude ASTC. Positive angles are measured in an anticlockwise direction. All Stations To Central Negative angles are measured in a clockwise direction.

4 For the signs of the ratios Ratios for Quadrant Boundary Angles Examining a point (x, y) on the unit circle: sin = y-coordinate cos = x-coordinate y-coordinate tan = x-coordinate Graphs of the Trigonometric Ratios y = cosec x 2. 1. 0 y = sin x 2 7 /4 3 /2 5 /4 3 /4 /2 /4 /4 /2 3 /4 5 /4 3 /2 7 /4 2 . 0. 1. 2. y = sin x y = cosec x Domain All real x 1 y 1. Range All reals except 0 , 180 , 360 etc. y 1, y 1. Period 360 360 .. Page 3 of 5. Trigonometry ( Mathematics & Mathematics Extension 1). Copyright Enoch Lau 2003 Jan 2003 (re-pub Apr 2003). 2. y = sec x 1. 0. y = cos x 2 7 /4 3 /2 5 /4 3 /4 /2 /4 /4 /2 3 /4 5 /4 3 /2 7 /4 2 . 0. 1. 2. y = cos x y = sec x Domain All real x 1 y 1. Range All reals except 90 , 270 , 450 etc.

5 Y 1, y 1. Period 360 360 . 2. 1. 0. 2 7 /4 3 /2 5 /4 3 /4 /2 /4 /4 /2 3 /4 5 /4 3 /2 7 /4 2 . 0 y = tan x 1. y = cot x 2. y = tan x y = cot x Domain All reals except 90 , 270 etc. All reals except 0 , 180 , 270 etc. Range All reals All reals Period 180 180 .. Page 4 of 5. Trigonometry ( Mathematics & Mathematics Extension 1). Copyright Enoch Lau 2003 Jan 2003 (re-pub Apr 2003). Trigonometric Identities sin . tan = cos . cos . cot = sin . sin 2 + cos 2 = 1. sin 2 = 1 cos 2 . cos 2 = 1 sin 2 . Dividing all terms by cos2 , Dividing all terms by sin2 , tan 2 + 1 = sec 2 1 + cot2 = cosec2 . 1 = sec 2 tan 2 1 = cosec2 cot2 . tan 2 = sec 2 1 cot2 = cosec2 1. 3-Dimensional Trigonometry ( Extension 1 only).

6 A straight line is perpendicular to a plane if it is perpendicular to every straight line that lies in the plane and passes through its foot. The perpendicular is called the normal to the plane at that point. The angle between a line and a plane is the angle between the line and its projection on the plane. The angle between two planes is measured by the angle between two lines drawn from any point in the line of intersection at right angles to it, one to each plane. The angle between two planes is called the dihedral angle. The line of greatest slope is the line that is perpendicular to the horizon.. Page 5 of 5.