Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CHAPTER 6: The Trigonometric Functions 6.1 6.2 6.3 6.4 6.5 6.6 The Trigonometric Functions of Acute Angles Applications of Right Triangles Trigonometric Functions of Any Angle Radians, Arc Length, and Angular Speed Circular functions: Graphs and Properties Graphs of Transformed Sine and Cosine Functions Copyright © 2009 Pearson Education, Inc. 6.3 Trigonometric Functions of Any Angle Find angles that are coterminal with a given angle and find the complement and the supplement of a given angle. Determine the six trigonometric function values for any angle in standard position when the coordinates of a point on the terminal side are given. Find the function values for any angle whose terminal side lies on an axis. Find the function values for an angle whose terminal side makes an angle of 30º, 45º, or 60º with the x-axis. Use a calculator to find function values and angles. Copyright © 2009 Pearson Education, Inc. Angle An angle is the union of two rays with a common endpoint called the vertex. We can think of it as a rotation. Locate a ray along the positive x-axis with its endpoint at the origin. This ray is called the initial side of the angle. Now rotate a copy of this ray. A rotation counterclockwise is a positive rotation and rotation clockwise is a negative rotation. The ray at the end of the rotation is called the terminal side of the angle. The angle formed is said to be in standard position. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 4 Angle Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 5 Angle The measure of an angle or rotation may be given in degrees. One complete positive revolution or rotation has a measure of 360º. One half of a revolution has a measure of 180º … Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 6 Angle One fourth of a revolution has a measure of 90º, and so on. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 7 Angle Angle measure of 60º, 135º, 330º, and 420º have terminal sides that lie in quadrants I, II, IV and I respectively. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 8 Angle The negative rotations –30º, –110º, and –225º represent angles with terminal sides in quadrants IV, III, and II respectively. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 9 Coterminal Angles If two or more angles have the same terminal side, the angles are said to be coterminal. To find angles coterminal with given angles, we add or subtract multiples of 360º. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 10 Example Find two positive angles and two negative angles that are coterminal with (a) 51º (b) –7º. Solution: a) Add or subtract multiples of 360º. Many answers are possible. 51º + 360º = 411º Copyright © 2009 Pearson Education, Inc. 51º + 3(360º) = 1131º Slide 6.3 - 11 Example Solution continued 51º – 360º = –309º 51º – 2(360º) = –669º b) We have the following: –7º + 360º = 353º –7º + 2(360º) = 713º –7º – 360º = –367º –7º – 10(360º) = –3607º Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 12 Classification of Angles Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 13 Complementary Angles Two acute angles are complementary if their sum is 90º. For example, angles that measure 10º and 80º are complementary because 10º + 80º = 90º. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 14 Supplementary Angles Two positive angles are supplementary if their sum is 180º. For example, angles that measure 45º and 135º are supplementary because 45º + 135º = 180º. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 15 Example Find the complement and supplement of 71.46º. Solution: 90º 71.46º 18.54º 180 71.46º 108.54º The complement of 71.46º is 18.54º and the supplement of 71.46º is 108.54º. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 16 Trigonometric Functions of Angles Consider a right triangle with one vertex at the origin of a coordinate system and one vertex on the positive x-axis. The other vertex P, a point on the circle whose center is at the origin and whose radius r is the length of the hypotenuse of the triangle. This triangle is a reference triangle for angle , which is in standard position. Note that y is the length of the side opposite and x is the length of the side adjacent to . Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 17 Trigonometric Functions of Angles The three trigonometric functions of are defined as follows: opp y adj x opp y sin cos tan hyp r hyp r adj x Since x and y are the coordinates of the point P and the length of the radius is the hypotenuse, we have: y-coordinate sin radius x-coordinate cos radius Copyright © 2009 Pearson Education, Inc. y-coordinate tan x-coordinate Slide 6.3 - 18 Trigonometric Functions of Angles We will use these definitions for functions of angles of any measure. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 19 Trigonometric Functions of Any Angle Suppose that P(x, y) is any point other than the vertex on the terminal side of any angle in standard position, and r is the radius, or distance from the origin to P(x,y). Then the trigonometric functions are defined as follows: y-coordinate y sin radius r radius r csc y-coordinate y x-coordinate x cos radius r y-coordinate y tan x-coordinate x radius sec x-coordinate x-coordinate cot y-coordinate Copyright © 2009 Pearson Education, Inc. r x x y Slide 6.3 - 20 Example Find the six trigonometric function values for the angle shown. Solution: Determine r, distance from (0, 0) to (–4, –3). r 2 2 x 0 y 0 x y r 4 3 2 2 Copyright © 2009 Pearson Education, Inc. 2 2 16 9 25 5 Slide 6.3 - 21 Example Solution continued Substitute –4 for x, –3 for y, and 5 for r. y 3 3 sin r 5 5 r 5 5 csc y 3 3 x 4 4 cos r 5 5 r 5 5 sec x 4 4 y 3 3 tan x 4 4 x 4 4 cot y 3 3 Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 22 Example 2 Given that tan and is in the second quadrant, 3 find the other function values. Solution: y 2 2 Sketch a second-quadrant angle using tan x 3 3 hyp 2 2 32 13 Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 23 Example Solution continued Use the lengths of the three sides to find the appropriate ratios. 2 2 13 sin 13 13 13 csc 2 3 3 13 cos 13 13 13 sec 3 2 tan 3 3 cot 2 Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 24 Terminal Side on an Axis An angle whose terminal side falls on one of the axes is a quadrantal angle. One of the coordinates of any point on that side is 0. The definitions of the trigonometric functions still apply, but in some cases, function values will not be defined because a denominator will be 0. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 25 Example Find the sine, cosine, and tangent values for 90º, 180º, 270º, and 360º. Solution: Sketch the angle in standard position, label a point on the terminal side, choosing (0, 1). 1 sin 90º 1 1 0 cos 90º 0 1 1 tan 90º Not defined 0 Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 26 Example Solution continued Copyright © 2009 Pearson Education, Inc. 0 sin180º 0 1 1 cos180º 1 1 0 tan180º 0 1 1 sin 270º 1 1 0 cos 270º 0 1 1 tan 270º Not defined 0 Slide 6.3 - 27 Example Solution continued 0 sin 360º 0 1 1 cos 360º 1 1 0 tan 360º 0 1 Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 28 Reference Angles: 30º (45º and 60º) Consider the angle 150º, its terminal side makes a 30º angle with the x-axis. 1 sin150º sin 30º 2 3 cos150º cos 30º 2 Copyright © 2009 Pearson Education, Inc. tan150º tan 30º 1 3 3 3 Slide 6.3 - 29 Example Find the sine, cosine, and tangent values for each of the following: a) 225º b) –780º Solution: Draw the figure, terminal side 225º, reference angle is 225º – 180º = 45º Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 30 Example Solution continued 2 sin 225º 2 2 cos 225º 2 tan 225º 1 Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 31 Example Solution continued Draw the figure, terminal side –780º is coterminal with –780º + 2(360º) = –60º, reference angle is 60º. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 32 Example Solution continued 3 sin 780º 2 1 cos 780º 2 tan 780º 3 Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 33 Example Given the function value and the quadrant restriction, find . a) sin = 0.2812, 90º < < 180º b) cot = –0.1611, 270º < < 360º Solution: Sketch the angle in the second quadrant. Use a calculator to find the acute (reference) angle whose sine is 0.2812. It’s approximately 16.33º. Now 180º – 16.33º = 163.37º. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 34 Example Solution continued b) cot = –0.1611, 270º < < 360º Sketch the angle in the fourth quadrant. 1 1 tan 6.2073 cot 0.1611 Use a calculator to find the acute (reference) angle whose tangent is –6.2073. It’s approximately 80.85º. Now 360º – 80.85 = 279.15º. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 35 Bearing: Second-Type In aerial navigation, directions, or bearings, are given in degrees clockwise from north. Thus east is 90º, south is 180º, and west is 270º. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 36 Example An airplane flies 218 mi from an airport in a direction of 245º. How far south of the airport is the plane then? How far west? Solution: Sketch a diagram. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 37 Example Solution continued: Find the measure of angle ABC: B 270º 245º 25º Find how far south the plane is, that is, the length b: b sin 25º 218 Copyright © 2009 Pearson Education, Inc. b 218sin 25º 92 mi Slide 6.3 - 38 Example Solution continued Find how far west the plane is, that is, the length a: a cos 25º 218 a 218 cos25º 198 mi The airplane is about 92 mi south and about 198 mi west of the airport. Copyright © 2009 Pearson Education, Inc. Slide 6.3 - 39