The six trigonometric functions

Transcription

1 PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 4: Trigonomic Functions 4.: Trigonomic Functions of Acute Angles What you'll Learn About Right Triangle Trigonometry/ Two Famous Triangles Evaluating Trig Functions with a calculator/applications of right triangle trig The six trigonometric functions 1 P a g e

2 P a g e Find the values of all six trigonometric functions.

3 Assume that is an acute angle in a right triangle satisfying the given conditions. Evaluate the remaining trigonometric functions. A) sin 4 9 B) cos 9 C) tan 4 9 D) cot 9 E) csc 10 7 F) sec P a g e

4 4 P a g e Triangle

5 5 P a g e Triangle

6 Evaluate using a calculator. Make sure your calculator is in the correct mode. Give answers to 3 decimal places and then draw the triangle that represents the situation. A) sin 53 B) cos 5 C) tan 154 D) cot 9 E) csc 0 F) 8 sec 5 6 P a g e

7 Solve the triangle for the variable shown. Solve the triangle ABC for all of its unknown parts. Assume C is the right angle. 40 a = 10 7 P a g e

8 Solve the triangle ABC for all of its unknown parts. Assume C is the right angle. 6 a = 7 Example 6: From a point 340 feet away from the base of the Peachtree Center Plaza in Atlanta, Georgia, the angle of elevation to the top of the building is 65. Find the height of the building. 8 P a g e

9 9 P a g e

10 PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 4: Trigonometric Functions 4.3: The circular functions What you'll Learn About Trig functions of any angle/trig functions of real numbers Periodic Functions/The Unit Circle Point P is on the terminal side of angle. Evaluate the six trigometric functions for. A) (5, 4) B) (-3, 4) C) (-, -5) D) (-4, -1) E) (0, -3) F) (3, 0) 10 P a g e

11 Determine the sign (+ or -) of the given value without the use of a calculator. A) sin 53 B) cos 5 C) tan 154 D) cot 9 E) csc 0 F) 8 sec 5 11 P a g e

12 1 P a g e

13 Evaluate without using a calculator by using ratios in a reference triangle. A) sin 10 B) cos 3 C) 13 tan 4 D) 13 cot 6 E) 7 csc F) 4 3 sec 6 13 P a g e

14 Find sine, cosine, and tangent for the given angle. A) 90 B) C) 6 D) 7 Evaluate without using a calculator A) Find sin and tan if cos 3 4 and cot 0 B) Find sec and csc if cot 6 5 and sin 0 14 P a g e

15 15 P a g e

16 PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 4: Trigonometric Functions 4.4: Graphs of sine and cosine What you'll Learn About The basic waves revisited/sinusoids and Transformations Modeling The graph of y = sin x The graph of y = cos x 16 P a g e

17 Find the amplitude of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = sinx 3 A) y = 3sinx B) y = sin x 4 C) y = - 5sinx Find the period of the function and use the language of transformations to describe how the graph of the function is related to the graph of y = cosx A) y = cos(x) B) cos x 3x y C) y cos 4 17 P a g e

18 Graph 1 period of the function without using your calculator. A) x y 3sin y 5cos x Identify the maximum and minimum values and the zeros of the function in the interval [, ]. Use your understanding of transformations, not your calculator. A) y = 4 sin x B) x y cos 3 18 P a g e

19 Determine the phase shift for the function and the sketch the graph. A) y cos x B) y sin x 6 3 Determine the vertical shift for the function and the sketch the graph. A) y cos x B) y sin x 3 Determine the vertical shift and phase shift of the function and then sketch the graph A) y cos x 1 B) y sin x P a g e

20 State the Amplitude and period of the sinusoid, and relative to the basic function, the phase shift and vertical translation. A) y 3sin x B) y cos 3x C) y 5sin 4 x 6 0 P a g e

21 Max Min Amp A Max Min Vertical (C) period p Horizontal Stretch/Shrink B p Construct a sinusoid with the given amplitude and period that goes through the given point. A) Amp: 4, period 4, point (0, 0) How to choose an appropriate model based on the behavior at some given time, T. y = A cos B(t T) + C if at time T the function attains a maximum value B) Amp:.5, period 5, point (, 0) y = -A cos B(t T) + C if at time T the function attains a minimum value y = A sin B(t T) + C if at time T the function halfway between a minimum and a maximum value y = -A sin B(t T) + C if at time T the function halfway between a maximum and a minimum value 1 P a g e

22 Max Min Amp A Max Min Vertical (C) period p Horizontal Stretch/Shrink B p How to choose an appropriate model based on the behavior at some given time, T. Example 7: Calculating the Ebb and Flow of Tides One particular July 4 th in Galveston, TX, high tide occurred at 9:36 am. At that time the water at the end of the 61 st Street Pier was.7 meters deep. Low tide occurred at 3:48 p.m, at which time the water was only.1 meters deep. Assume that the depth of the water is a sinusoidal function of time with a period of half a lunar day (about 1 hrs 4 min) a) Model the depth, D, as a sinusoidal function of time, t, algebraically then graph the function. y = A cos B(t T) + C if at time T the function attains a maximum value y = -A cos B(t T) + C if at time T the function attains a minimum value y = A sin B(t T) + C if at time T the function halfway between a minimum and a maximum value y = -A sin B(t T) + C if at time T the function halfway between a maximum and a minimum value b) At what time on the 4 th of July did the first low tide occur. c) What was the approximate depth of the water at 6:00 am and at 3:00 pm? d) What was the first time on July 4 th when the water was.4 meters deep? P a g e

23 80) Temperature Data: The normal monthly Fahrenheit temperatures in Helena, MT, are shown in the table below (month 1 = January) Model the temperature T as a sinusoidal function of time using 0 as the minimum value and 68 as the maximum value. Support your answer graphically by graphing your function with a scatter plot. M T P a g e

24 4 P a g e

25 PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 4: Trigonometric Functions 4.5: Graphs of Tan/Cot/Sec/Csc What you'll Learn About The graphs of the other 4 trig functions The graph of y = csc x The graph of y = sec x 5 P a g e

26 The graph of y = tan x The graph of y = cot x 6 P a g e

27 Describe the graph of the function in terms of a basic trigonometric function. Locate the vertical asymptotes and graph periods of the function. A) y = tan(3x) B) y = -cot(x) C) y = sec(4x) D) y csc x 3 7 P a g e

28 Describe the transformations required to obtain the graph of the given function form a basic trigonometric graph. A) y 5tan x x B) y 3cot C) 4x y sec D) y 4csc x P a g e

29 9 P a g e

30 PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 4: Trigonometric Fucntions 4.7: Inverse Trigonometric Functions What you'll Learn About Inverse Trigonometric Functions and their Graphs The graph of y = sin x The graph of y sin 1 x arcsin x 30 P a g e

31 31 P a g e The Unit Circle and Inverse Functions

32 The graph of y = cos x The graph of y cos 1 x arccos x The Unit Circle and Inverse Functions 3 P a g e

33 The graph of y = tan x The graph of y tan 1 x arctan x 33 P a g e

34 Find the exact value A ) cos B ) cos C ) cos D ) sin E ) sin F ) sin -1-1 G ) tan 1 ) tan H I ) tan 3-1 J ) cos 0 ) sin -1-1 K 1 L ) tan 0 34 P a g e

35 Use a calculator to find the approximate value in degrees. Draw the triangle that represents the situation. A ) arccos(.456) B ) arcsin.456 C ) arctan Use a calculator to find the approximate value in radians. Draw the triangle that represents the situation. A ) arcsin(.456) B ) arccos.456 C ) arctan P a g e

36 Find the exact value without a calculator. 1 1 A ) sin cos 1 B ) cos tan 0 1 C ) D ) sintan 3 1 tansin -1-1 E ) cos sin F ) sin cos P a g e

37 37 P a g e

38 PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Solving Trigonometric Equations What you'll Learn About Solve each trigonometric equation for on the interval 0,. Then give a formula for all possible angles that could be a solution of the equation. A) sin B) 1 cos C) sin 1 D) cos 0 E) tan 3 F) tan 1 38 P a g e

39 Solve each trigonometric equation for on the interval 0,. A) cos 1 B) sin 3 1 C) cos 3 3 D) tan 1 3 E) sin. 4 F) cos. 39 P a g e

40 A) cos 1 0 B) 3 csc 0 C) 4sin 1 0 D) 3cot 1 cot 3 0 E) 3tan 1 0 F) cos 3sin 40 P a g e

41 G) cos cos 0 H) sin cos cos I) csc csc J) sin 3 sin 41 P a g e

42 4 P a g e

43 PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 5: Analytic Trigonometry 5.1 Fundamental Identities Reciprocal Identities 1 1 csc x secx sin x cos x 1 1 sin x cosx csc x sec x 1 cotx tan x 1 tanx cot x tan x Quotient Identities sin x cos x cotx cos x sin x Even/Odd Identities sin(-x) = -sin x cos(-x) = cos x tan(-x) = -tan x csc(-x) = -csc x sec(-x) = sec x cot(-x) = -cot x Pythagorean Identities sin x + cos x = 1 tan x + 1 = sec x 1 + cot x = csc x sin x = 1 - cos x tan x = sec x 1 csc x - cot x = 1 cos x = 1 - sin x sec x tan x = 1 cot x = csc x - 1 Co-function sin x cos x cos x sin x tan x cot x csc x sec x sec x csc x cot x tan x 43 P a g e

44 Equation of Unit Circle x y 1 Use trig ratios to prove that cos sin 1 is the equation of the unit circle Use basic identities to simplify the expression to a different trig function or a product of two trig functions 1 sin 10. cot xtan x A. cos 3 B. sin x sin x C. sin x cot x cos csc x x 44 P a g e

45 Simplify the expression to either 1 or -1 x 19. cot xcot x 17. sin csc( x) 1. sin x cos ( x) Simplify the expression to either a constant or a basic trig function. cot xsec x 1 cot x A) B) sec x 1 tan x C) tan x cot x sec x csc x 45 P a g e

46 Use the basic identities to change the expression to one involving only sines and cosines. Then simplify to a basic trig function. 8) sin tan cos cos sec y tan y sec y tan y 30) sec y 31. tan x tan x csc x sec x 46 P a g e

47 Combine the fractions and simplify to a multiple of a power of a basic trig function 1 1 A) 1 cos x 1 cos x 35. sin x cot x sin x cos x Write each expression in factored form as an algebraic expression of a single trig function A) sin x sin x 1 B) 1 cos x cos x 4 C) sin x cos x tan x sin xcsc x cot x 47 P a g e

48 Write each expression as an algebraic expression of a single trigonometric function 1 cos x cot 1 A) 1 cos x B) 1 cot x C) cos x 1 sin x D) cot x 1 csc x 48 P a g e

49 PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 5: Analytic Trigonometry 5. Proving Trig Identities What you'll Learn About cotx cosxtanx cos x sin x 1. sinx 14. cosx sinx 1 sin xcos x 18. sec 1 sin sin 1 sin 49 P a g e

50 csc 1- cosx 1 cos x x. sin cos 1 cos secx 1 6. tanx sinx 1- cosx 50 P a g e

51 30. t an sin tan sin 35. tanx secx -1 secx 1 tanx 37. sinx - cosx sin x 1 sinx cosx 1 sinxcosx 51 P a g e

52 5 P a g e

53 53 P a g e