Digital Lesson Right Triangle Trigonometry

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 opp hyp cos adj hyp tan opp adj csc hyp opp sec hyp adj cot adj opp Copyright by Houghton Mifflin Company, Inc. All rights reserved. 2

Copyright by Houghton Mifflin Company, Inc. All rights reserved. 3

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

Copyright by Houghton Mifflin Company, Inc. All rights reserved. 5

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. 6

Copyright by Houghton Mifflin Company, Inc. All rights reserved. 7

Calculate the trigonometric functions for a 60 angle. 2 3 60 opp 3 sin 60 hyp 2 1 1 adj cos 60 2 hyp 3 opp tan 60 3 1 adj 3 1 cot 60 adj 3 3 opp hyp 2 sec 60 2 adj 1 2 2 3 hyp csc 60 opp 3 3 Copyright by Houghton Mifflin Company, Inc. All rights reserved. 8

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 a sin and cos (90 ) . b b hyp 90 – θ a θ b 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 ) 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 10

Copyright by Houghton Mifflin Company, Inc. All rights reserved. 11

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. sin 15 4 1 4 15 tan 15 1 cos Copyright by Houghton Mifflin Company, Inc. All rights reserved. 15 θ 1 4 1 sin 15 1 sec 4 cos 1 cot 15 csc 12

Copyright by Houghton Mifflin Company, Inc. All rights reserved. 13