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Digital Lesson Right Triangle Trigonometry The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. The sides of the right triangle are: hyp the side opposite the acute angle , opp the side adjacent to the acute angle , θ and the hypotenuse of the right triangle. adj The trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. opp sin = cos = adj tan = opp hyp hyp adj csc = hyp opp sec = hyp adj Copyright © by Houghton Mifflin Company, Inc. All rights reserved. cot = adj opp 2 Calculate the trigonometric functions for . 5 4 3 The six trig ratios are 4 sin = 5 4 tan = 3 5 sec = 3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 cos = 5 3 cot = 4 5 csc = 4 3 Example adj. cos A hyp. x cos 55 .5735 9 x 9(0.5735) x 5.16 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Sines, Cosines, and Tangents of Special Angles sin 30 sin sin 45 sin sin 60 sin 6 4 3 1 3 3 , cos 30 cos , tan 30 tan 2 6 2 6 3 2 2 , cos 45 cos , tan 45 tan 1 2 4 2 4 3 1 , cos 60 cos , tan 60 tan 3 2 3 2 3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Trigonometric Identities are trigonometric equations that hold for all values of the variables. Example: sin = cos(90 ), for 0 < < 90 Note that and 90 are complementary angles. Side a is opposite θ and also adjacent to 90○– θ . hyp θ a a sin = and cos (90 ) = . hyp hyp So, sin = cos (90 ). Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 90○– θ a b 6 Fundamental Trigonometric Identities for 0 < < 90. Cofunction Identities sin = cos(90 ) tan = cot(90 ) sec = csc(90 ) cos = sin(90 ) cot = tan(90 ) csc = sec(90 ) Reciprocal Identities sin = 1/csc cot = 1/tan cos = 1/sec sec = 1/cos tan = 1/cot csc = 1/sin Quotient Identities tan = sin /cos cot = cos /sin Pythagorean Identities sin2 + cos2 = 1 tan2 + 1 = sec2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. cot2 + 1 = csc2 7 Example: Given sin = 0.25, find cos , tan , and sec . Draw a right triangle with acute angle , hypotenuse of length one, and opposite side of length 0.25. Use the Pythagorean Theorem to solve for the third side. 0.9682 = 0.9682 1 tan = 0.25 = 0.258 0.9682 1 sec = = 1.033 0.9682 cos = Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1 0.25 θ 0.9682 8 Example: Given sec = 4, find the values of the other five trigonometric functions of . Draw a right triangle with an angle such 4 4 hyp that 4 = sec = = . adj 1 Use the Pythagorean Theorem to solve for the third side of the triangle. 15 4 cos = 1 4 tan = 15 = 15 1 sin = Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 θ 1 1 = 4 sin 15 1 sec = =4 cos 1 cot = 15 csc = 9 Angle of Elevation and Angle of Depression When an observer is looking upward, the angle formed by a horizontal line and the line of sight is called the: angle of elevation. line of sight object angle of elevation horizontal observer When an observer is looking downward, the angle formed by a horizontal line and the line of sight is called the: horizontal angle of depression line of sight object observer angle of depression. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Example 2: A ship at sea is sighted by an observer at the edge of a cliff 42 m high. The angle of depression to the ship is 16. What is the distance from the ship to the base of the cliff? observer cliff 42 m horizontal 16○ angle of depression line of sight 16○ d ship 42 = 146.47. tan 16 The ship is 146 m from the base of the cliff. d= Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Text Example Sighting the top of a building, a surveyor measured the angle of elevation to be 22º. The transit is 5 feet above the ground and 300 feet from the building. Find the building’s height. Solution Let a be the height of the portion of the building that lies above the transit in the figure shown. The height of the building is the transit’s height, 5 feet, plus a. Thus, we need to identify a trigonometric function that will make it possible to find a. In terms of the 22º angle, we are looking for the side opposite the angle. a Transit 5 feet 22º Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 300 feet h Text Example cont. Solution The transit is 300 feet from the building, so the side adjacent to the 22º angle is 300 feet. Because we have a known angle, an unknown opposite side, and a known adjacent side, we select the tangent function. a tan 22º = 300 Length of side opposite the 22º angle Length of side adjacent to the 22º angle a = 300 tan 22º 300(0.4040) 121 Multiply both sides of the equation by 300. The height of the part of the building above the transit is approximately 121 feet. If we add the height of the transit, 5 feet, the building’s height is approximately 126 feet. a Transit 5 feet 22º Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 300 feet h Geometry of the 45-45-90 triangle Consider an isosceles right triangle with two sides of length 1. 45 2 1 12 12 2 45 1 The Pythagorean Theorem implies that the hypotenuse is of length 2 . Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Calculate the trigonometric functions for a 45 angle. 2 1 45 1 sin 45 = opp 1 2 = = hyp 2 2 adj 1 cot 45 = = = 1 opp 1 opp 1 tan 45 = = = 1 1 adj sec 45 = 2 hyp = = 1 adj 1 2 adj cos 45 = = = 2 hyp 2 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. csc 45 = 2 hyp = = 2 opp 1 15 Geometry of the 30-60-90 triangle Consider an equilateral triangle with each side of length 2. 30○ 30○ The three sides are equal, so the angles are equal; each is 60. 2 The perpendicular bisector of the base bisects the opposite angle. 60○ 2 3 1 60○ 2 1 Use the Pythagorean Theorem to find the length of the altitude, 3 . Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Calculate the trigonometric functions for a 30 angle. 2 1 30 3 opp 1 sin 30 = = hyp 2 cos 30 = 3 1 opp tan 30 = = = adj 3 3 3 adj cot 30 = = = 3 1 opp 2 2 3 hyp sec 30 = = = 3 3 adj hyp 2 csc 30 = = = 2 opp 1 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 adj = 2 hyp 17 Calculate the trigonometric functions for a 60 angle. 2 3 60○ opp 3 sin 60 = = hyp 2 tan 60 = 1 3 opp = = 3 1 adj hyp 2 sec 60 = = = 2 adj 1 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1 adj cos 60 = = 2 hyp 3 1 cot 60 = adj = = opp 3 3 csc 60 = 2 2 3 hyp = = opp 3 3 18