Right Triangle Trigonometry Digital Lesson

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

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Calculate the trigonometric functions for . The six trig ratios are  sin  = tan  = sec  = cos  = cot  = csc  = Example: Six Trig Ratios

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

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Sines, Cosines, and Tangents of Special Angles

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 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. a hyp b θ 90 ○ – θ sin  = and cos (90    ) =. So, sin  = cos (90    ). Example: Using Trigonometric Identities

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 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  cos  = 1/sec  tan  = 1/cot  cot  = 1/tan  sec  = 1/cos  csc  = 1/sin  Quotient Identities tan  = sin  /cos  cot  = cos  /sin  Pythagorean Identities sin 2  + cos 2  = 1 tan 2  + 1 = sec 2  cot 2  + 1 = csc 2  Fundamental Trigonometric Identities for

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Given sin  = 0.25, find cos , tan , and sec . Example: Given 1 trig function, find other functions cos  = = tan  = = sec  = = Use the Pythagorean Theorem to solve for the third side Draw a right triangle with acute angle , hypotenuse of length one, and opposite side of length θ

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Example: Given sec  = 4, find the values of the other five trigonometric functions of . Use the Pythagorean Theorem to solve for the third side of the triangle. tan  = = cot  =sin  = csc  = = cos  = sec  = = 4 θ 4 1 Draw a right triangle with an angle  such that 4 = sec  = =. adj hyp Example: Given 1 Trig Function, Find Other Functions

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 angle of elevation When an observer is looking downward, the angle formed by a horizontal line and the line of sight is called the: Angle of Elevation and Angle of Depression When an observer is looking upward, angle of elevation. the angle formed by a horizontal line and the line of sight is called the: observer object line of sight horizontal observer object line of sight horizontal angle of depression angle of depression.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 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? The ship is 146 m from the base of the cliff. line of sight angle of depression horizontal observer ship cliff 42 m 16 ○ d Example 2: Application d = =

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 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. 22º Line of sight 300 feet a 5 feet h Transit Text Example

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Transit 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. Solution 22º Line of sight 300 feet a 5 feet h tan 22º = a 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 = 300 tan 22º 300(0.4040) 121 Multiply both sides of the equation by 300. Length of side opposite the 22º angleLength of side adjacent to the 22º angle Text Example cont.

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

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

Copyright © by Houghton Mifflin Company, Inc. All rights reserved ○ Consider an equilateral triangle with each side of length 2. The perpendicular bisector of the base bisects the opposite angle. The three sides are equal, so the angles are equal; each is 60 . Geometry of the triangle ○ Use the Pythagorean Theorem to find the length of the altitude,. Geometry of the Triangle

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

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