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Pre-AP/GT Pre-Calculus Assignment Sheet Unit 7 – Inverse Trig Functions January 19th – 25th, 2017 Thursday 1/19 Graphing Inverse Functions Pages 6-7 Evaluating Trig functions Page 10 Monday 1/23 Evaluating Trig functions Quiz on Inverse Functions Page 12 Tuesday 1/24 Solving Trig. Equations with Calculator and by hand Page 14 Friday 1/20 Review Wednesday 1/25 Study! Unit Test 1 Notes on Graphing Inverse Trig Functions Let’s start by graphing Sine, Cosine, and Tangent below: Sine Cosine Tangent Are these functions considered One-To-One? Why or why not? If a function is not One-To-One, their domains must be restricted (this will allow their inverses to be a function). How to find the inverse of a Trig Function: Function Sketch Inverse Domain Range Notice: When we found the Domain and Ranges for the inverse functions, we switched the functions Domain and Range. This comes from Algebra II when you first learned to find the inverse (switch x and y). Next, let’s sketch the graphs of the inverses. 2 We can use the domain and ranges to sketch the graphs of the inverse functions y arcsin( x) or y sin 1 ( x) y arccos( x) or y cos 1 ( x) , Domain: 1, 1 Range: 2 2 Domain: 1, 1 Range: 0, y arctan( x) or y tan 1 ( x) , Domain: , Range: 2 2 Having restricted the interval on which we graph so that each inverse is a function results in only one answer for each problem. The range of sine and tangent is in Quadrants I and IV, while the range of cosine is Quadrants I and II. Label this information on a coordinate plane below. 3 Graph the following Inverse Trig Functions: State the Domain and Range. 1. y sin 1 (2 x) 2. y arccos( x) 4. y cos 1 (2 x 2) x 5. y arctan( ) 2 2 3. y arcsin( x 1) 6. y sin 1 ( x 2) 2 4 Assignment on Graphing Inverse Trig functions Show all work. Graphing Inverse Trig Equations: Sketch a graph of each of the following and state the domain and range 1. y = sin-1(3x) 2. y = 3. y = arcsin(x + 1) 4. y = 2sin-1(x) 5. y = arccos(2x – 4) 6. y = tan-1(x) + π 5 Graphing Inverse Trig Equations: Sketch a graph of each of the following and state the domain and range x 7. y arcsin( ) 2 9. Let sin x 8. y 2 cos 1 ( x 2) 5 . Find the exact values of all six trig functions. (Hint…Draw a Right Triangle) 7 10. Solve the following for x: a. sin 1 x 4 b. arccos( x) 5 6 c. tan 1 x 0 6 Notes on Inverse Functions (Day 1) Review of Domain Restrictions and Quadrants: arccos x arcsin x arctan x Inverse Trig Functions - Draw a reference triangle and evaluate each of the following expressions. Remember to be careful of which quadrant. 1 1. sin arccos 2 5 3. tan arccos 6 3 2. sin arccos 5 4. cos arc csc 13 5 7 Find the exact values without using a calculator. 1. sin 1 1 = 3. sin 1 1.5 = 2. Arc csc2 = 1 2 4. A rccos = 5. A rc cot 1 = 3 Find the exact value or angle in terms of . Remember to look at the outside function to determine if the answer will be an angle or value. 1 2 1. sin A rccos 4. sin 1 cos 5 4 4 2. Arc sec sec 3 4 5. sin cos 1 3. tan 1 sin 5 1 2 6. tan sin 1 8 Assignment on Inverse Functions (day 1) Show all work. Find the exact values without using a calculator. 2 3 2. sec1 2 1. csc 1 3. cos 1 3 2 Inverse Trig Functions - Draw a reference triangle and evaluate each of the following expressions. Remember to be careful of which quadrant. 15 5 1. cos sin1 17 1 3 9. tan arcsin 5 3 13 3. sin cos1 3 12 6. sec cot 1 5 7. tan sec1 12 5. cot tan1 10 2. sin cos1 13 6 4. sin csc1 5 15 8. csc tan1 8 1 10. cos arcsin 4 Find the exact value or angle in terms of . Remember to look at the outside function to determine if the answer will be an angle or value. 11. A rcsin sin 7 6 12. sin 1 sin 2 7 16. sin tan 1(1) 15. cos A rcsin 3 2 13. cos 1 cot 4 5 13 17. cos A rcsin 2 9 14. csc cot 1 1 18. csc cos 1 9 Notes on Inverse Trig Functions (Day 2) Use an Inverse Trigonometric to write as a function of x. 1. 2. Properties of Inverse Trig Functions sin(arcsin x) x and sin(arcsin y ) y iff 1 x 1 and cos(arccos x) x and cos(arccos y ) y iff 1 x 1 and 0 x tan(arctan x) x and tan(arctan y ) y iff 1 x 1 and 2 2 x x 2 2 Evaluate the following: 1. sin(arcsin 0.6) 2. tan(arctan 35) Write an Algebraic Expression that is equivalent to the expression. 1. sin(arctan x) x 3 3. cot(arccos ) 2. sec(arctan 3x) 4. sec(arcsin ( x 1)) 10 Notes on Solving Trig Equations with Calculators Determine the values of , where 0 360 , to the nearest hundredth of a degree. Before you begin: Make sure you are in DEGREE MODE!!!!!! On your calculator. Determine the reference angle using your calculator. Where could the angle lie? Quadrant I, II, II, IV Find both angle values of . 1. sin = .7183 2. tan = 1.6198 3. cos = – .6691 4. sec = – 4.8097 (2nd cos 1/– 4.8097 ) 5. cot = – .1228 (2nd tan 1/– .1228) Determine the values of , where 0 2 , to the nearest hundredth of a radian. Before you begin: Make sure you are in RADIAN MODE!!!!!! On your calculator. 6. sin = – .8183 7. tan = 2.4567 8. csc = – 1.1859 Fun Ones: Solve the following: 9. 3 sec 12 sec 21 10. 3 cos2 2 cos 1 0 11 Assignment on Solving Trig Equations with Calculators Determine the values of , where 0 360 , to the nearest hundredth of a degree. 1. sin = 0.4067 2. cos = – 0.5023 3. tan = 2.9988 4. sec = 1.1111 5. cot = – 1.2222 6. csc = 2.5012 Determine the values of , where 0 2 , to the nearest hundredth of a radian. (Radian Mode) 7. sin = 0.8143 8. cos = 0.7838 9. tan = –.2677 10. csc = 1.0204 11. cot = 0.5890 12. sec = – 1.5861 Solve each of the following on the interval from [0, 2π) 13. 11csc x + 15 = 9csc x + 19 14. 2cos2x – 1 = 0 15. 2 cos 2 x 3 0 16. 2 sin2 x 5 sin x 3 0 12 Notes on Solving Trig Equations by Hand Solve the following equations on the interval 0 2 . Give the exact answer in terms of . 1. 2sin 1 0 4. 2 csc 2 0 7. 2 sin 2 sin 0 3 cot 1 0 2. 4 tan x 4 0 3. 5. tan 3 2 tan 6. sec 2 2 0 8. 2 sin 2 sin 1 0 13 Assignment on Solving Trig Equations by Hand Solve the following equations on the interval 0 2 . Give the exact answer in terms of . 1. tan 3 0 3 csc 2 0 2. 2 cos 3 0 3. 5. 5sec 10 0 6. 4 cos 2 1 7. cos 2 3cos 8. tan 2 tan 0 9. 2 cos 2 5 cos 2 0 10. sec 2 0 11. 5sin 13 3sin 14 12. 4. 2 cos 1 0 3 cot 1 0 Extra Credit: Solve the following equations on the interval 0 2 . Give the exact answer in terms of . tan sec tan 0 14