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Trigonometry Copyright 2011 Pearson Canada Inc. T-1 §1 Angles and Radian Measure Copyright 2011 Pearson Canada Inc. T-2 Angles A ray is a part of a line that has only one endpoint and extends forever in the opposite direction. A rotating ray is often a useful means of thinking about angles. An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other the terminal side. Copyright 2011 Pearson Canada Inc. T-3 Common Angles Copyright 2011 Pearson Canada Inc. T-4 Measuring Angles Using Radians A central angle is angle whose vertex is at the centre of a circle. Central angle Intercepted arc An intercepted arc is the distance along the circumference of the circle between the initial and terminal side of a central angle. Copyright 2011 Pearson Canada Inc. T-5 One-Radian Angle If the length of the intercepted arc is equal to the circle’s radius, then we say the central angle measures one radian. For 1-radian angle, the intercepted arc and the radius are equal. Copyright 2011 Pearson Canada Inc. T-6 Radian Measure Angles measured in radians. Copyright 2011 Pearson Canada Inc. T-7 Radian Measure Let θ be a central angle in a circle of radius r and let s be the length of its intercepted arc. The measure of θ is: s radians. r Copyright 2011 Pearson Canada Inc. T-8 Radian Measure Example: A central angle, θ , in a circle of radius 5 centimetres intercepts an arc of length 20 centimetres. What is the radian measure of θ? s 20 cm 4 r 5cm 20 cm The radian measure of θ is 4. 5 cm Copyright 2011 Pearson Canada Inc. T-9 Converting Between Degrees and Radians The measure of one complete rotation in radians is: s 2 r 2 radians r r The measure of one complete rotation is also 360˚, so 360˚ = 2π radians. Dividing both sides by 2 gives: 180˚ = π radians Copyright 2011 Pearson Canada Inc. T - 10 Converting Between Degrees and Radians Conversion Between Degrees and Radians Using the basic relationship π radians = 180˚, 1. To convert degrees to radians, multiply degrees by radians 180 2. To convert radians to degrees, multiply radians by 180 radians Copyright 2011 Pearson Canada Inc. T - 11 Converting Between Degrees and Radians Example: Convert each angle in degrees to radians. 135˚ 120˚ 135 135 120 120 Copyright 2011 Pearson Canada Inc. radians 180 radians 180 135 3 radians radians 180 4 120 2 radians radians 180 3 T - 12 Converting Between Degrees and Radians Example: Convert each angle in radians to degrees. 5 6 3 5 5 radians 180 5 180 radians 150 6 6 radians 6 3 radians Copyright 2011 Pearson Canada Inc. radians 3 180 180 60 radians 3 T - 13 §2 Angles and the Cartesian Plane Copyright 2011 Pearson Canada Inc. T - 14 Drawing Angles in Standard Position An angle is in standard position on the xy-plane if its vertex is at the origin and its initial side lies along the positive x-axis. y x Copyright 2011 Pearson Canada Inc. T - 15 Drawing Angles in Standard Position A positive angle is generated by a counterclockwise rotation form the initial side to the terminal side. A negative angle is generated by a clockwise rotation form the initial side to the terminal side. Copyright 2011 Pearson Canada Inc. T - 16 Drawing Angles in Standard Position y The xy-plane is divided into four quadrants. Quadrant II Quadrant I x Quadrant III Quadrant IV If the terminal side of the angle lies on the x-axis or y-axis the angle is called a quadrantal angle. Copyright 2011 Pearson Canada Inc. T - 17 Angles Formed by Revolution of Terminal Sides Copyright 2011 Pearson Canada Inc. 18 Drawing Angles in Standard Position Example: Draw and label each angle in standard position. y Terminal side 3 Vertex Copyright 2011 Pearson Canada Inc. 3 Initial side x T - 19 Drawing Angles in Standard Position Example: Draw and label each angle in standard position. y 2 Vertex Initial side x Terminal side Copyright 2011 Pearson Canada Inc. 2 T - 20 Degree and Radian Measures of Common Angles Copyright 2011 Pearson Canada Inc. T - 21 Coterminal Angles Two angles with the same initial and terminal side but possibly different rotations are called coterminal angles. Coterminal Angles Measured in Degrees An angle of θ˚ (an angle measured in degrees) is coterminal with angles of θ˚ + 360˚k, where k is an integer. Two coterminal angles for an angle of θ˚ can be found by adding 360˚ to θ˚ and subtracting 360˚ from θ˚. Copyright 2011 Pearson Canada Inc. 22 Coterminal Angles Copyright 2011 Pearson Canada Inc. 23 Coterminal Angles Example: Assume the following angle is in standard position. Find a positive angle less than 360˚ that is coterminal with it. 460˚ 460˚ – 360˚ = 100˚ Angles of 460˚ and 100˚ are coterminal. Copyright 2011 Pearson Canada Inc. T - 24 Coterminal Angles Example: Assume the following angle is in standard position. Find a positive angle less than 360˚ that is coterminal with it. – 60˚ – 60˚ + 360˚ = 300˚ Angles of – 60˚ and 300˚ are coterminal. Copyright 2011 Pearson Canada Inc. T - 25 Coterminal Angles Coterminal Angles Measured in Radians An angle of θ radians (an angle measured in radians) is coterminal with angles of θ + 2πk, where k is an integer. Copyright 2011 Pearson Canada Inc. 26 Coterminal Angles Example: Assume the following angle is in standard position. Find a positive angle less than 2π that is coterminal with it. 7 2 7 7 4 3 2 2 2 2 2 Angles of 7 and 3 are coterminal. 2 2 Copyright 2011 Pearson Canada Inc. T - 27 §3 Right Triangle Trigonometry Copyright 2011 Pearson Canada Inc. T - 28 Labelling a Right Triangle Using the standard labelling of a right triangle, we label its sides and angles so that side a is opposite to angle A, side b is opposite to angle B, and side c is opposite to angle C. Hypotenuse Leg Leg Angle C is always taken to be the right angle, making side c the hypotenuse. Copyright 2011 Pearson Canada Inc. T - 29 The Pythagorean Theorem The Pythagorean Theorem in terms of the standard labelling of a right triangle is given by c a b 2 2 2 Hypotenuse Leg Leg Copyright 2011 Pearson Canada Inc. T - 30 The Pythagorean Theorem Example: Find the length of the hypotenuse c where a = 3 cm and b = 4 cm. Hypotenuse c a b 2 2 c 3 4 2 c 25 2 2 2 2 a=3 cm b=4 cm c 25 5 The length of the hypotenuse is 5 cm. Copyright 2011 Pearson Canada Inc. T - 31 Primary Trigonometric Ratios The three primary trigonometric ratios and their abbreviations are Name Abbreviation Sine sin Cosine cos Tangent tan Consider a right triangle with one of its acute angles labelled θ. Copyright 2011 Pearson Canada Inc. T - 32 Primary Trigonometric Ratios Right Triangle Definitions of Sine, Cosine, and Tangent The three primary trigonometric ratios of the acute angle θ are defined as follows: length of side opposite angle a sin length of hypotenuse c length of side adjacent to angle b cos length of hypotenuse c tan length of side opposite angle a length of side adjacent to angle b Copyright 2011 Pearson Canada Inc. T - 33 Primary Trigonometric Ratios Trigonometry values for a given angle are always the same no matter how large the triangle is. Copyright 2011 Pearson Canada Inc. T - 34 Primary Trigonometric Ratios Example: Find the value of each of the three primary trigonometric ratios of θ. c a b 2 2 2 (2 5) a 4 2 2 2 c2 5 b4 20 a 16 2 a 4 a2 2 Example continues. Copyright 2011 Pearson Canada Inc. T - 35 Primary Trigonometric Ratios Example: Find the value of each of the three primary trigonometric ratios of θ. c2 5 opposite 2 sin hypotenuse 2 5 adjacent 4 cos hypotenuse 2 5 opposite 2 1 tan adjacent 4 2 Copyright 2011 Pearson Canada Inc. 1 5 2 5 b4 Example continues. T - 36 Primary Trigonometric Ratios of Special Angles 2 sin 45 2 2 cos 45 2 tan 45 1 Copyright 2011 Pearson Canada Inc. T - 37 Primary Trigonometric Ratios of Special Angles 1 sin 30 2 3 sin 60 2 3 cos 30 2 3 tan 30 3 1 cos 60 2 Copyright 2011 Pearson Canada Inc. tan 60 3 T - 38 Primary Trigonometric Ratios Using a Calculator Example: Use a calculator to find the value to four decimal places. sin cos 1.2 3 Function Mode cos 1.2 Radian sin 3 Radian Copyright 2011 Pearson Canada Inc. Display, rounded to four decimal places Keystrokes COS 1.2 SIN ( 0.3624 = π ÷ 3 ) = 0.8660 T - 39 §4 Solving Applied Problems Involving Trigonometry Copyright 2011 Pearson Canada Inc. T - 40 Solving Right Triangles Solving a right triangle means finding the missing lengths of its sides and the measurements of its angles. Copyright 2011 Pearson Canada Inc. T - 41 Solving Right Triangles Example: Solve the given triangle, rounding lengths to two decimal places. B 90 A 90 40 50 a tan 40 12 a 12 tan 40 10.07 Copyright 2011 Pearson Canada Inc. 40˚ 12 c c cos 40 12 12 c 15.66 cos 40 cos 40 T - 42 Applied Problems An angle formed by a horizontal line and the line of sight to an object that is above the horizontal line is called the angle of elevation. Copyright 2011 Pearson Canada Inc. T - 43 Applied Problems The angle formed by a horizontal line and the line of sight to an object that is below the horizontal line is called the angle of depression. Copyright 2011 Pearson Canada Inc. T - 44 Applied Problems Example: The irregular blue shape is a pond. The distance across the pond, a, is unknown. To find this distance a surveyor took the measurements shown in the figure. What is the distance across the pond? a sin 24 1200 a 1200sin 24 488 1200 m The distance across the pond is 488 m. Copyright 2011 Pearson Canada Inc. T - 45 Applied Problems Example: A building is 40 metres high and it casts a shadow 36 metres long. Find the angle of elevation of the sun to the nearest degree. 40 tan 36 1 40 tan 48 36 The angle of elevation is 48˚. Copyright 2011 Pearson Canada Inc. 40m 36m T - 46