Digital Lesson Right Triangle Trigonometry The six trigonometric

  • Slides: 13
Download presentation
Digital Lesson Right Triangle Trigonometry

Digital Lesson Right Triangle Trigonometry

The six trigonometric functions of a right triangle, with an acute angle , are

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

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

Geometry of the 45 -45 -90 triangle Consider an isosceles right triangle with two

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. 4

Calculate the trigonometric functions for a 45 angle. 1 45 1 = adj cos

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. 5

Geometry of the 30 -60 -90 triangle Consider an equilateral triangle with each side

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. 6

Calculate the trigonometric functions for a 30 angle. 2 1 30 sin 30 =

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 7

Calculate the trigonometric functions for a 60 angle. 2 60○ 1 sin 60 =

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. 8

Trigonometric Identities are trigonometric equations that hold for all values of the variables. Example:

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. 9

Fundamental Trigonometric Identities for 0 < < 90. Cofunction Identities sin = cos(90 )

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 10

Example: Given sin = 0. 25, find cos , tan , and sec .

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. 11

Example: Given sec = 4, find the values of the other five trigonometric functions

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 = 12

Example: Given sin = 0. 25, find cos , tan , and sec .

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. 13