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Right Triangle Trigonometry Digital Lesson

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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

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

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Geometry of the 45-45-90 triangle Consider an isosceles right triangle with two sides of length 1. 1 1 45 The Pythagorean Theorem implies that the hypotenuse is of length. Geometry of the 45-45-90 Triangle

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Calculate the trigonometric functions for a 45 angle. Examp le: Trig Functio ns for 45 1 1 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

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 60 ○ 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 30-60-90 triangle 22 2 11 30 ○ Use the Pythagorean Theorem to find the length of the altitude,. Geometry of the 30-60-90 Triangle

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Calculate the trigonometric functions for a 30 angle. 1 2 30 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 = =

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Calculate the trigonometric functions for a 60 angle. 1 2 60 ○ 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 = =

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 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

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 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

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example: Given sin = 0.25, find cos , tan , and sec . Example: Given 1 trig function, find other functions cos = = 0.9682 tan = = 0.258 sec = = 1.033 Use the Pythagorean Theorem to solve for the third side. 0.9682 Draw a right triangle with acute angle , hypotenuse of length one, and opposite side of length 0.25. θ 1 0.25

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 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

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Example: Given sin = 0.25, find cos , tan , and sec . Example: Given 1 trig function, find other functions cos = = 0.9682 tan = = 0.258 sec = = 1.033 Use the Pythagorean Theorem to solve for the third side. 0.9682 Draw a right triangle with acute angle , hypotenuse of length one, and opposite side of length 0.25. θ 1 0.25

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