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PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 4: Trigonomic Functions 4.2: 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|Page Find the values of all six trigonometric functions. 2|Page Assume that is an acute angle in a right triangle satisfying the given conditions. Evaluate the remaining trigonometric functions. 3|Page A) sin 4 9 C) tan 4 9 D) cot E) csc 10 7 F) sec B) cos 2 9 2 9 4 3 45-45-90 Triangle 30-60-90 Triangle 4|Page 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 C) tan154 E) csc 220 5|Page B) cos D) cot F) sec 2 5 9 8 5 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 6|Page Solve the triangle ABC for all of its unknown parts. Assume C is the right angle. 62 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. 7|Page 8|Page PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 4: Trigonometric Functions 4.2: 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) (-2, -5) E) (0, -3) 9|Page D) (-4, -1) F) (3, 0) Determine the sign (+ or -) of the given value without the use of a calculator. A) sin 53 C) tan154 E) csc 220 10 | P a g e B) cos D) cot F) sec 2 5 9 8 5 11 | P a g e Evaluate without using a calculator by using ratios in a reference triangle. 12 | P a g e A) sin 120 B) cos C) tan 13 4 D) cot E) csc 7 4 F) sec 2 3 13 6 23 6 Find sine, cosine, and tangent for the given angle. A) 90 B) 2 D) 5 C) 6 Evaluate without using a calculator 13 | P a g e A) Find sin and tan if cos 3 and cot 0 4 B) Find sec and csc if cot 6 and sin 0 5 14 | P a g e PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 4: Trigonometric Functions 4.3: 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 15 | P a g e 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 A) y = 3sinx B) y = 3 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(2x) 16 | P a g e B) y cos x 2 3x C) y cos 4 Graph 3 periods of the function without using your calculator. A) y 3 sin x 2 y 5 cos 2 x Identify the maximum and minimum values and the zeros of the function in the interval [2 ,2 ] . Use your understanding of transformations, not your calculator. A) y = 4 sin x 17 | P a g e B) y 2 cos x 3 Construct a sinusoid with the given amplitude and period that goes through the given point. A) Amp: 4, period 4 , point (0, 0) B) Amp: 2.5, period , point (2, 0) 5 State the Amplitude and period of the sinusoid, and relative to the basic function, the phase shift and vertical translation. A) y 3 sin x 2 4 C) y 5 sin 4x 6 18 | P a g e B) y 2 sin 3x 4 4 Max Min 2 M ax M in Vertical Shift 2 period p Amp Horizontal Stretch/Sh rink B 2 p Example 7: Calculating the Ebb and Flow of Tides One particular July 4th in Galveston, TX, high tide occurred at 9:36 am. At that time the water at the end of the 61st Street Pier was 2.7 meters deep. Low tide occurred at 3:48 p.m, at which time the water was only 2.1 meters deep. Assume that the depth of the water is a sinusoidal function of time with a period of half a lunar day (about 12 hrs 24 min) a) Model the depth, D, as a sinusoidal function of time, t, algebraically then graph the function. b) At what time on the 4th of July id 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 4th when the water was 2.4 meters deep? 19 | P a g e 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 20 as the minimum value and 68 as the maximum value Support your answer graphically by graphing your function with a scatter plot. M T 20 | P a g e 1 20 2 26 3 35 4 44 5 53 6 61 7 68 8 67 9 56 10 45 11 31 12 21 21 | P a g e 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 22 | P a g e The graph of y = tan x The graph of y = cot x 23 | P a g e Describe the graph of the function in terms of a basic trigonometric function. Locate the vertical asymptotes and graph 2 periods of the function. A) y = 2tan(3x) C) y = sec(4x) 24 | P a g e B) y = -cot(2x) D) y csc 3x Describe the transformations required to obtain the graph of the given function form a basic trigonometric graph. 25 | P a g e A) y 5 tan x C) y 2 sec 4x 3 B) y 3 cot 2x D) y 4 csc 2x 3 26 | P a g e 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 1 x arcsin x The graph of y cos 1 x arccos x The graph of y tan1 x arctan x 27 | P a g e The Unit Circle and Inverse Functions 28 | P a g e Find the exact value A) cos -1 3 2 D) sin -1 3 2 G) tan-1 1 J ) cos -1 0 29 | P a g e B) cos -1 1 C ) cos -1 2 1 2 E ) sin -1 1 F ) sin -1 2 1 2 H ) tan-1 3 K ) sin -1 1 1 I ) tan-1 3 L) tan-1 0 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- 5.768 Use a calculator to find the approximate value in radians. Draw the triangle that represents the situation. A) arcsin (.456) C ) arctan- 5.768 30 | P a g e B) arccos .456 Find the exact value without a calculator. A) sin cos 1 1 2 31 | P a g e B) cos tan1 0 2 C ) tan sin 1 2 D) sin tan1 3 E ) cos -1 sin 4 F ) sin -1 cos 6 32 | P a g e PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Solving Trigonometric Equations What you'll Learn About Solve each trigonometric equation for on the interval 0,2 . A) sin 2 2 C) sin 1 E) cos 2 33 | P a g e B) cos 1 2 D) cos 0 1 2 F) sin 3 1 2 G) cos 3 I) sin .4 34 | P a g e 3 2 H) tan 1 2 3 35 | P a g e A) 2 sin 1 0 C) 4 cos 2 1 0 E) 2 cos 2 cos 0 B) 3 sec 2 0 D) 3 tan2 1tan2 3 0 F) 2 sin cos cos 36 | P a g e PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 5: Analytic Trigonometry 5.1 Fundamental Identities Reciprocal Identity Example csc 4 1 sin 4 Even/Odd Identity Example cos 37 | P a g e cos 4 4 Quotient Identity Example tan 4 sin cos 4 4 Cofunction Identity Example sin90 60 cos(60) Equation of Unit Circle x2 y 2 1 2 2 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 2 10. cot x tan x A. cos B. sin x sin x 3 38 | P a g e C. sin 2 x cot 2 x cos 2 x csc x Simplify the expression to either 1 or -1 17. sin x csc( x) 19. cot x cot x 2 2 2 21. sin x cos ( x) Simplify the expression to either a constant or a basic trig function. cot x sec x 1 cot x 2 A) B) 2 1 tan x sec x 2 2 2 2 C) tan x cot x sec x csc x 39 | P a g e Use the basic identities to change the expression to one involving only sines and cosines. Then simplify to a basic trig function. 28) sin tan cos cos 2 30) 31. 40 | P a g e sec y tan y sec y tan y sec y tan x tan x 2 csc x sec 2 x 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 sin x 2 cot x cos 2 x Write each expression in factored form as an algebraic expression of a single trig function A) sin 2 x 2 sin x 1 C) sin x 2 cos 2 x 1 41 | P a g e B) 1 2 cos x cos 2 x 45. 4 tan2 x 4 sin x csc x cot x Write each expression as an algebraic expression of a single trigonometric function A) 1 cos 2 x 1 cos x cos 2 x C) 1 sin x cot 2 1 1 cot x B) D) cot 2 x 1 csc x Find all solutions to the equation in the interval [0, 2π] A) 2cosxsinx + sinx = 0 C) cos x tan2 x cos x 42 | P a g e B) 2 cot x sin x cot x 0 D) cot 2 x 3 Find all solutions to the equation in the interval [0, 2π] A) 4 sin 2 x 4 sin x 1 0 B) cos 2 x 2 cos x 0 C) 3 cos x 2 sin 2 x D) 2 cos 2 x 3 cos x 2 Find all solutions to the equation in the interval [0, 2π] A) cosx = .53 43 | P a g e B) sinx = .75 C) cos2x=.9 44 | P a g e PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 5: Analytic Trigonometry 5.2 Proving Trig Identities What you'll Learn About 12. sinxcotx cosxtanx cos x sin 2 x 14. cosx sinx 1 2 sin xcos x 2 18. 45 | P a g e sec2 1 sin sin 1 sin 2 20. 1 1 2 csc2 x 1 - cosx 1 cos x 22. sin 2 cos 2 1 2cos 2 26. 46 | P a g e secx 1 sinx tanx 1 - cosx 47 | P a g e 30. tan 2 sin 2 tan2 sin 2 35. tanx secx 1 secx - 1 tanx 37. sinx - cosx 2 sin 2 x 1 sinx cosx 1 2sinxcosx 48 | P a g e 49 | P a g e