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 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. 5 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. 7 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. 8 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  = =

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

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

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  = = 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. 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

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  = = 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