1. 4.3 Fundamental Trig Identities and Right Triangle Trig Applications 2015 Trig Identities Sheet Handout

2. Homework Quiz 4.2 10/30/13 Evaluate the trigonometric function using its period as an aid. Exact value only. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

3. 3 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

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

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

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 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

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

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 1 2 30 Example: Trig Functions for  30  60 ○ sin 30 ○ = cos 30 ○ = tan 30 ○ = cot 30 ○ = sec 30 ○ = csc 30 ○ = cos 60 ○ = sin 60 ○ = cot 60 ○ = tan 60 ○ = csc 60 ○ = sec 60 ○ =

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 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  = = 60 ○

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

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

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

13. Negative Angle Identities Remember: if f(-t) = f(t) the function is even if f(-t) = - f(t) the function is odd The cosine and secant functions are EVEN. cos(-t)=cos t sec(-t)=sec t The sine, cosecant, tangent, and cotangent functions are ODD. sin(-t)= -sin tcsc(-t)= -csc t tan(-t)= -tan tcot(-t)= -cot t (1, 0)(–1, 0) (0,–1) (0,1) x y x

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 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

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Example: Given sin  = 2/5, find the values of the other five trigonometric functions of . Example: Given 1 trig function, find other functions tan  = cot  = cos  =sec  = csc  = θ

16. Use trigonometric identities to find the indicated trigonometric functions. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16

17. Use trigonometric identities to transform one side of the equation into the other. a. b. c. d. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17

18. Applying Trig You are 200 yards from a river. Rather than walking directly to the river, you walk 400 yards, diagonally, along a straight path to the rivers edge. Find the acute angle between this path and the river’s edge. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18

19. Applying Trig A surveyor is standing 50 feet from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as How tall is the tree? Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19

20. Homework 4.3 p 274 5-15 odd, 27-43 odd, 47-55 odd Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20

21. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21 There is another way to state the size of an angle, one that subdivides a degree into smaller pieces. In a full circle there are 360 degrees. Each degree can be divided into 60 parts, each part being 1/60 of a degree. These parts are called minutes. Each minute can divided into 60 parts, each part being 1/60 of a minute. These parts are called seconds. Degrees, Minutes, and Seconds

22. Degrees, Minutes, and Seconds Conversions To convert decimal degrees into DMS, multiply decimal degrees by 60 To convert from DMS to decimal degrees, divide minutes by 60, seconds by 3600 OR use the Angle feature of your calculator

23. Examples