Digital Lesson Right Triangle Trigonometry The six trigonometric

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 , the side adjacent to the acute angle , θ and the hypotenuse of the right triangle. opp adj The trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. sin = cos =adj hyp tan =opp adj hyp csc = opp sec =hyp adj cot =adj opp Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

Calculate the trigonometric functions for . 5 4 3 The six trig ratios are sin = cos = tan = cot = sec = csc = Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3

Example Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4

Sines, Cosines, and Tangents of Special Angles 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. hyp Side a is opposite θ and also adjacent to 90○– θ a θ b sin = and cos (90 ) = . So, sin = cos (90 ). Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6

Fundamental Trigonometric Identities for 0 < < 90. Cofunction Identities sin = cos(90 ) cos = sin(90 ) tan = cot(90 ) cot = tan(90 ) sec = csc(90 ) csc = sec(90 ) Reciprocal Identities sin = 1/csc cot = 1/tan cos = 1/sec = 1/cos tan = 1/cot csc = 1/sin Quotient Identities tan = sin /cos cot = cos /sin Pythagorean Identities sin 2 + cos 2 = 1 tan 2 + 1 = sec 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. cot 2 + 1 = csc 2 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. cos = = 0. 9682 tan = = 0. 258 sec = 1 θ 0. 25 0. 9682 = 1. 033 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8

Example: Given sec = 4, find the values of the other five trigonometric functions of . Draw a right triangle with an angle such that 4 = sec = hyp = adj 4 . Use the Pythagorean Theorem to solve for the third side of the triangle. θ 1 sin = csc = = cos = sec = =4 tan = = Copyright © by Houghton Mifflin Company, Inc. All rights reserved. cot = 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: angle of depression. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. horizontal angle of depression line of sight object observer 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 d= ship = 146. 47. The ship is 146 m from the base of the cliff. 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. Transit Lin f eo 22º Copyright © by Houghton Mifflin Company, Inc. All rights reserved. a ht sig 5 feet 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 a = 300 tan 22º 300(0. 4040) 121 Length of side adjacent to the 22º angle 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. Transit Lin f eo 22º Copyright © by Houghton Mifflin Company, Inc. All rights reserved. a ht sig 5 feet 300 feet h

Geometry of the 45 -45 -90 triangle Consider an isosceles right triangle with two sides of length 1. 45 1 The Pythagorean Theorem implies that the hypotenuse is of length. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14

Calculate the trigonometric functions for a 45 angle. 1 45 1 = adj cos 45 = = hyp = opp tan 45 = = adj = 1 adj cot 45 = = opp = 1 hyp sec 45 = = adj = hyp csc 45 = = opp = sin 45 = = Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15

Geometry of the 30 -60 -90 triangle Consider an equilateral triangle with each side of length 2. 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 1 Use the Pythagorean Theorem to find the length of the altitude, . Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16

Calculate the trigonometric functions for a 30 angle. 2 1 30 sin 30 = adj cos 30 = = hyp = opp tan 30 = = adj hyp sec 30 = = adj = opp = cot 30 = = hyp csc 30 = = opp Copyright © by Houghton Mifflin Company, Inc. All rights reserved. = = 2 17

Calculate the trigonometric functions for a 60 angle. 2 60○ 1 sin 60 = tan 60 = = opp = adj hyp sec 60 = = adj cos 60 = = hyp = cot 60 = adj = opp = = 2 hyp csc 60 = = opp = Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18