Copyright  2011 Pearson Canada Inc. Trigonometry T - 1

Copyright  2011 Pearson Canada Inc. Angles and Radian Measure § 1 T - 2

Copyright  2011 Pearson Canada Inc. 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. T - 3

Copyright  2011 Pearson Canada Inc. Common Angles T - 4

Copyright  2011 Pearson Canada Inc. Measuring Angles Using Radians A central angle is angle whose vertex is at the centre of a circle. An intercepted arc is the distance along the circumference of the circle between the initial and terminal side of a central angle. Intercepted arc Central angle T - 5

Copyright  2011 Pearson Canada Inc. 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. T - 6

Copyright  2011 Pearson Canada Inc. Radian Measure Angles measured in radians. T - 7

Copyright  2011 Pearson Canada Inc. 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: T - 8

Copyright  2011 Pearson Canada Inc. 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? The radian measure of θ is 4. 5 cm 20 cm T - 9

Copyright  2011 Pearson Canada Inc. Converting Between Degrees and Radians The measure of one complete rotation in radians is: The measure of one complete rotation is also 360˚, so 360˚ = 2π radians. Dividing both sides by 2 gives: 180˚ = π radians T - 10

Copyright  2011 Pearson Canada Inc. 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 2.To convert radians to degrees, multiply radians by T - 11

Copyright  2011 Pearson Canada Inc. Converting Between Degrees and Radians Example: Convert each angle in degrees to radians. 135˚120˚ T - 12

Copyright  2011 Pearson Canada Inc. Converting Between Degrees and Radians Example: Convert each angle in radians to degrees. T - 13

Copyright  2011 Pearson Canada Inc. Angles and the Cartesian Plane § 2 T - 14

Copyright  2011 Pearson Canada Inc. 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. x y T - 15

Copyright  2011 Pearson Canada Inc. 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. T - 16

Copyright  2011 Pearson Canada Inc. Drawing Angles in Standard Position The xy-plane is divided into four quadrants. T - 17 y Quadrant I x Quadrant II Quadrant IIIQuadrant 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. Angles Formed by Revolution of Terminal Sides 18

Copyright  2011 Pearson Canada Inc. Drawing Angles in Standard Position Example: Draw and label each angle in standard position. T - 19 y x Terminal side Initial side Vertex

Copyright  2011 Pearson Canada Inc. Drawing Angles in Standard Position Example: Draw and label each angle in standard position. T - 20 y x Terminal side Initial side Vertex

Copyright  2011 Pearson Canada Inc. Degree and Radian Measures of Common Angles T - 21

Copyright  2011 Pearson Canada Inc. Coterminal Angles 22 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. Coterminal Angles 23

Copyright  2011 Pearson Canada Inc. Coterminal Angles Example: Assume the following angle is in standard position. Find a positive angle less than 360˚ that is coterminal with it. 460˚ T ˚ – 360˚ = 100˚ Angles of 460˚ and 100˚ are coterminal.

Copyright  2011 Pearson Canada Inc. Coterminal Angles Example: Assume the following angle is in standard position. Find a positive angle less than 360˚ that is coterminal with it. – 60˚ T - 25 – 60˚ + 360˚ = 300˚ Angles of – 60˚ and 300˚ are coterminal.

Copyright  2011 Pearson Canada Inc. Coterminal Angles 26 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. Coterminal Angles Example: Assume the following angle is in standard position. Find a positive angle less than 2π that is coterminal with it. T - 27 Angles of and are coterminal.

Copyright  2011 Pearson Canada Inc. Right Triangle Trigonometry § 3 T - 28

Copyright  2011 Pearson Canada Inc. Leg Hypotenuse 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. T - 29 Angle C is always taken to be the right angle, making side c the hypotenuse.

Copyright  2011 Pearson Canada Inc. Leg Hypotenuse The Pythagorean Theorem The Pythagorean Theorem in terms of the standard labelling of a right triangle is given by T - 30

Copyright  2011 Pearson Canada Inc. Hypotenuse a=3 cm b=4 cm The Pythagorean Theorem Example: Find the length of the hypotenuse c where a = 3 cm and b = 4 cm. T - 31 The length of the hypotenuse is 5 cm.

Copyright  2011 Pearson Canada Inc. Primary Trigonometric Ratios The three primary trigonometric ratios and their abbreviations are NameAbbreviation Sinesin Cosinecos Tangenttan Consider a right triangle with one of its acute angles labelled θ. T - 32

Copyright  2011 Pearson Canada Inc. 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: T - 33

Copyright  2011 Pearson Canada Inc. Primary Trigonometric Ratios Trigonometry values for a given angle are always the same no matter how large the triangle is. T - 34

Copyright  2011 Pearson Canada Inc. Primary Trigonometric Ratios Example: Find the value of each of the three primary trigonometric ratios of θ. Example continues. T - 35

Copyright  2011 Pearson Canada Inc. Primary Trigonometric Ratios Example: Find the value of each of the three primary trigonometric ratios of θ. Example continues. T - 36

Copyright  2011 Pearson Canada Inc. Primary Trigonometric Ratios of Special Angles T - 37

Copyright  2011 Pearson Canada Inc. Primary Trigonometric Ratios of Special Angles T - 38

Copyright  2011 Pearson Canada Inc. Primary Trigonometric Ratios Using a Calculator Example: Use a calculator to find the value to four decimal places. cos 1.2 FunctionModeKeystrokes Display, rounded to four decimal places cos 1.2Radian COS 1.2 = SIN (π ÷) = 3 T

Copyright  2011 Pearson Canada Inc. Solving Applied Problems Involving Trigonometry § 4 T - 40

Copyright  2011 Pearson Canada Inc. Solving Right Triangles Solving a right triangle means finding the missing lengths of its sides and the measurements of its angles. T - 41

Copyright  2011 Pearson Canada Inc. 40˚ Solving Right Triangles Example: Solve the given triangle, rounding lengths to two decimal places. T - 42

Copyright  2011 Pearson Canada Inc. 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. T - 43

Copyright  2011 Pearson Canada Inc. 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. T - 44

Copyright  2011 Pearson Canada Inc. 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? The distance across the pond is 488 m. T - 45

Copyright  2011 Pearson Canada Inc. 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. The angle of elevation is 48˚. 40m 36m T - 46