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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. 6.1 Trigonometric Functions of Acute Angles Determine the six trigonometric ratios for a given acute angle of a right triangle. Determine the trigonometric function values of 30º, 45º, and 60º. Using a calculator, find function values for any acute angle, and given a function value of an acute angle, find the angle. Given the function values of an acute angle, find the function values of its complement. Copyright © 2009 Pearson Education, Inc. Right Triangles and Acute Angles An acute angle is an angle with measure greater than 0º and less than 90º. Greek letters such as (alpha), (beta), (gamma), (theta), and (phi) are often used to denote an angle. We label the sides with respect to angles. The hypotenuse is opposite the right angle. There is the side opposite and the side adjacent to . Hypotenuse Side opposite Side adjacent to Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 4 Trigonometric Ratios The lengths of the sides of a right triangle are used to define the six trigonometric ratios: sine (sin) cosine (cos) tangent (tan) cosecant (csc) secant (sec) cotangent (cot) Hypotenuse Side opposite Side adjacent to Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 5 Trigonometric Function Values of an Acute Angle Let be an acute angle of a right triangle. Then the six trigonometric functions of are as follows: side opposite sin hypotenuse hypotenuse csc side opposite side adjacent to cos hypotenuse hypotenuse sec side adjacent to side opposite tan side adjacent to side adjacent to cot side opposite Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 6 Example In the triangle shown, find the six trigonometric function values of (a) and (b) . 12 13 Solution: opp 12 a) sin hyp 13 5 adj 5 cos hyp 13 hyp 13 sec adj 5 opp 12 tan adj 5 adj 5 cot opp 12 hyp 13 csc opp 12 Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 7 Example In the triangle shown, find the six trigonometric function values of (a) and (b) . 12 13 Solution: opp 5 a) sin hyp 13 5 adj 12 cos hyp 13 hyp 13 sec adj 12 opp 5 tan adj 12 adj 12 cot opp 5 hyp 13 csc opp 5 Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 8 Reciprocal Functions Note that there is a reciprocal relationship between pairs of the trigonometric functions. 1 csc sin 1 sec cos 1 cot tan Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 9 Example 4 3 4 Given that sin , cos , and tan , 5 5 3 find csc , sec , and cot . Solution: 1 4 5 5 4 1 1 sec 3 cos 5 5 3 1 csc sin Copyright © 2009 Pearson Education, Inc. 1 cot tan 1 3 4 4 3 Slide 6.1 - 10 Example 6 If sin and is an acute angle, find the other 7 five trigonometric function values of . Solution: Use the definition of the sine function that the ratio 6 opp and draw a right triangle. 7 hyp Use the Pythagorean equation to find a. 7 6 a a2 b2 c2 a2 62 72 a2 36 49 Copyright © 2009 Pearson Education, Inc. a2 49 36 13 a 13 Slide 6.1 - 11 Example Solution continued Use the lengths of the three sides to find the other five ratios. 6 sin 7 7 csc 6 13 cos 7 7 7 13 sec 13 13 6 6 13 tan 13 13 13 cot 6 Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 12 Function Values of 45º A right triangle with one 45º, must have a second 45º, making it an isosceles triangle, with legs the same length. Consider one with legs of length 1. opp 1 2 sin 45º 0.7071 hyp 2 2 45º 2 1 45º 1 adj 1 2 cos 45º 0.7071 hyp 2 2 opp 1 tan 45º 1 adj 1 Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 13 Function Values of 30º A right triangle with 30º and 60º acute angles is half an equilateral triangle. Consider an equilateral triangle with sides 2 and take half of it. 1 sin 30º 0.5, 2 2 30º 3 60º 1 Copyright © 2009 Pearson Education, Inc. 3 cos 30º 0.8660, 2 1 3 tan 30º 0.5774 3 3 Slide 6.1 - 14 Function Values of 60º A right triangle with 30º and 60º acute angles is half an equilateral triangle. Consider an equilateral triangle with sides 2 and take half of it. 3 sin 60º 0.8660, 2 2 30º 3 60º 1 Copyright © 2009 Pearson Education, Inc. 1 cos 60º 0.5, 2 3 tan 60º 3 1.7321 1 Slide 6.1 - 15 Example As a hot-air balloon began to rise, the ground crew drove 1.2 mi to an observation station. The initial observation from the station estimated the angle between the ground and the line of sight to the balloon to be 30º. Approximately how high was the balloon at that point? (We are assuming that the wind velocity was low and that the balloon rose vertically for the first few minutes.) Solution: Draw the situation, label the acute angle and length of the adjacent side. Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 16 Example Solution continued: opp h tan 30º adj 1.2 3 1.2 h 3 0.7 h 1.2 tan 30º h The balloon is approximately 0.7 mi, or 3696 ft, high. Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 17 Function Values of Any Acute Angle Angles are measured either in degrees, minutes, and seconds: 1º = 60´, 1´ = 60´´; referred to as the DºM´S´´ form 61 degrees, 27 minutes, 42 seconds 61º 2742 or are measured in decimal degree form, expressing the fraction parts of degrees in decimal form 61º 2742 61.45 1 Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 18 Examples Find the trigonometric function value, rounded to four decimal places, of each of the following: a) tan29.7º b) sec 48º c) sin 84º1039 Solution: Check that the calculator is in degree mode. a) tan29.7º 0.5703899297 0.5704 1 b) sec 48º 1.49447655 1.49445 cos 48º c) sin 84º1039 0.9948409474 0.9948 Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 19 Example A paint crew has purchased new 30-ft extension ladders. The manufacturer states that the safest placement on a wall is to extend the ladder to 25 ft and to position the base 6.5 ft from the wall. What angle does the ladder make with the ground in this position? Solution: Draw the situation, label the hypotenuse and length of the side adjacent to . Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 20 Example Solution continued: 6.5 ft adj cos 25 ft hyp 0.26 Use a calculator to find the acute angle whose cosine is 0.26: 74.92993786º Thus when the ladder is in its safest position, it makes an angle of about 75º with the ground. Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 21 Cofunction Identities Two angles are complementary whenever the sum of their measures is 90º. Here are some relationships. sin cos 90º 90º – cos sin 90º tan cot 90º cot tan 90º sec csc 90º csc sec 90º Copyright © 2009 Pearson Education, Inc. Slide 6.1 - 22 Example Given that sin 18º ≈ 0.3090, cos 18º ≈ 0.9511, and tan 18º ≈ 0.3249, find the six trigonometric function values of 72º. Solution: 1 csc18º 3.2361 sin18º sin 72º cos18º 0.9511 cos 72º sin18º 0.3090 1 sec18º 1.0515 cos18º tan 72º cot18º 3.0777 1 cot18º 3.0777 tan18º sec 72º csc18º 3.2361 Copyright © 2009 Pearson Education, Inc. cot 72º tan18º 0.3249 csc 72º sec18º 1.0515 Slide 6.1 - 23