17 Logarithmic Properties, Solving Exponential & Logarithmic

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1 Logarithmic Properties, Solving Eponential & Logarithmic Equations & Models Concepts: Properties of Logarithms Simplifying Logarithmic Epressions Proving the Quotient Rule for Logarithms Using the Change of Base Formula to Find Approimate Values of Logarithms Solving Eponential and Logarithmic Equations Algebraically Strategies: Same base eponential epressions that are equal must have equal eponents. Same base logarithmic epressions that are equal must have equal arguments. Isolate eponential epression, rewrite equation in logarithmic form. Isolate logarithmic epression, rewrite equation in eponential form. Solving Eponential and Logarithmic Models/Applications (Section.4 -.6). Prove the Quotient Rule for Logarithms. ( u ) log a = log v a (u) log a (v) Proof:

2 . Write each epression in terms of log(), log(y), and log(z) if possible. If it is not possible, eplain why. ( ) y (a) log z ( ) + y (b) log z (c) log ( yz ). Use your calculator to find approimate values for the following. (a) ln() (b) log() (c) log (6) (d) log () 4. Given the magnitude of an earthquake on the Richter scale is given by R(i) = log( i i 0 ), where i is the amplitude of the ground motion of the earthquake and i 0 is the amplitude of the ground motion of the zero earthquake, (a) Find the magnitude on the Richter scale of an earthquake that is 0000 times stronger than the zero quake. (b) Find the magnitude on the Richter scale of an earthquake that is times stronger than the zero quake.. The half-life of a certain radioactive substance is,6 years. (a) Find the decay rate constant r. (b) How much substance will be left in 00 years if there is currently 00 grams of the substance? 6. Find how long it takes for a deposit to double in value if the annual interest rate is.% and the interest is compounded continuously.. The antibiotic clarithromycin is eliminated from the body according to the formula A(t) = 00e 0.86t, where A is the amount remaining in the body (in milligrams) t hours after the drug reaches peak concentration. (a) How much time will pass before the amount of drug in the body is reduced to 00 milligrams? (b) Find the inverse of A(t) and eplain what the inverse function models.

3 8. You are given models for the population (in millions) of different countries t years after 00. For each part, determine the year in which the models predict the populations will be equal. (Source: World Health Organization s 006 World Health Statistics) (a) Rwanda: R(t) = 9.04(.0) t and Hungary: H(t) = 0.(0.98) t. (b) Cambodia: C(t) = 4.0(.0) t and Kazakhstan: K(t) = 4.8(0.9) t. 9. Find the solution(s) of the following eponential equations. Your answers should be eact. (a) 0 = 0 9 (b) + = 0 (c) + e = (d) 4 6 = 0 (e) 9e 8 = 0. Find the solution(s) of the following logarithmic equations. Your answers should be eact. (a) log 4 ( + ) + log 4 = log 4 + log 4 ( ) (b) log ( + ) log ( ) = (c) log (log ) = 4 (d) log( + ) = log + log (e) log 8 ( ) + log 8 ( + ) =. Suppose you re driving your car on a cold winter day (0 F outside) and the engine overheats (at about 0 F). When you park, the engine begins to cool down. The temperature U of the engine t minutes after you park satisfies the equation ( ) U 0 ln = 0.t. 00 (a) Solve the equation for U. (b) Use part (a) to find the temperature of the engine after 0 min (t = 0).. Joni invests $000 at an interest rate of % per year compounded continuously. How much time will it take for the value of the investment to quadruple.. Joni invests $000 at an interest rate of % per year compounded monthly. How much time will it take for the value of the investment to quadruple.

4 8 Angles and Their Measurement Concepts: Angles Initial Side and Terminal Side Standard Position Coterminal Angles Measuring Angles Radian Measure vs. Degree Measure Radian Measure as a Distance on the Unit Circle Converting between Radian Measure and Degree Measure Finding the Quadrant Associated with the Terminal Side of an Angle Identifying the Point on the Unit Circle that Corresponds to an Angle in Standard Position (Sections 6.). Find the radian measure of each of the following: (a) 40 angle (b) 0 angle. Show which of the following points must lie on the unit circle. (a) (0, ) (b) (, ) (c) (, 4 ) (d) ( 4, ) (e) (, ) (f) (, ) (g) (, ) (h) (, )

5 . ( Suppose than an angle of measure θ radians intersects the unit circle at the point ),. (a) What is one possibility for θ? (b) How do you find all the other possibilities? 4. Suppose than an angle of measure θ radians intersects the unit circle at the point ( ),. (a) What is one possibility for θ? (b) How do you find all the other possibilities?. Suppose that an angle of measure θ radians is placed in standard position. Find the location of the terminal side of the angle. Possibilities: (A) Quadrant I, (B) Quadrant II, (C) Quadrant III, (D) Quadrant VI, (E) the positive -ais, (F) the negative -ais, (G) the positive y-ais, or (H) the negative y-ais. (a) θ = 4π (b) θ = 4π (c) θ = 00π (d) θ = 00π (e) θ = π (f) θ = π (g) θ = 0π (h) θ = 0π 6. Find the terminal point on the unit circle determined by the given value of θ. (a) θ = 4π (b) θ = π (c) θ = π 6 (d) θ = π 6 (e) θ = π 4 (f) θ = π (g) θ = 4π (h) θ = π 6

6 9 Introduction to Trigonometry Worksheet Concepts: The Trigonometric Functions The Definitions of sin, cos, and tan Based on the Unit Circle Evaluating the Basic Trigonometric Functions at Special Angles The Sign of a Trigonometric Function The π π π or the Triangle The π π π or the Triangle Approimating Values of Trigonometric Functions with Your Calculator Parentheses Are Important Radian Mode vs. Degree Mode Understanding Trigonometric Notation The Graph of the sin, cos, and tan Functions Applying Graph Transformations to the Graphs of the sin, cos, and tan Functions Using Graphical Evidence to Make Conjectures about Identities (Sections 6. & 6.4). Evaluate the basic trigonometric functions at each of the following angles. (a) θ = π (b) θ = 9π 4 (c) θ = 4π (d) θ = π 6. (a) The terminal side of an angle, θ, in standard position contains the point (, 9). Evaluate the basic trigonometric functions at θ. (b) The terminal side of an angle, θ, in standard position contains the point (, 4). Evaluate the basic trigonometric functions at θ.. Suppose θ is in the fourth quadrant and cos(θ) =. Evaluate the remaining basic trigonometric functions on θ.

7 4. Suppose θ is in the first quadrant and sin(θ) = 6. Find each of the following (give eact answers): (a) sin(8π + θ) (b) tan( θ) (c) cos(4π θ). For each of the following equations, list the transformations that you need to apply to the graph of y = sin(), y = cos(), or y = tan() to sketch the graph of the equation. Sketch the graph. Be sure that the graph is well-labeled. (a) y = 4 sin () ( ) (b) y = cos π ( (c) y = sin π ) ( (d) y = tan + π ) 4 (e) y = u cos(v + w). (Assume that u, v, and w are positive.) 6. Use graphical evidence to determine which of the following MIGHT be trigonometric identities and which definitely cannot be trigonometric identities. (a) cos() = cos () sin () (b) sin() = sin () cos () (c) sin( + y) = sin() + sin(y) (d) sin() cos(y) = (sin( + y) + sin( y)) (e) tan(y) = tan() tan(y) (f) sin(t ) t = sin(t)

8 . The graph of a periodic function is shown below. Find a rule for the function. y π π π 4π 8. The graph of a periodic function is shown below. Find a rule for the function. y π π π 4π

9 9. (Question 4 from Section 6. of your tetbook) Burke s blood pressure can be modeled by the function g(t) = cos(.πt) +, where t is the time (in seconds) and f(t) is in millimeters of mercury. The highest pressure (systolic) occurs when the heart beats, and the lowest pressure (diastolic) occurs when the heart is at rest between beats. The blood pressure is the ratio systolic/diastolic. (a) Graph the blood pressure function over a period of two seconds and determine Burke s blood pressure. (b) Find Burke s pulse rate (number of heartbeats per minute). (c) According to current guidelines, someone with systolic pressure above 40 or diastolic pressure above 90 has high blood pressure and should see a doctor about it. What would you advise the person in this case? 0. (Question 4 from Section 6. of your tetbook) The current generated by an AM radio transmitter is given by a function of the form f(t) = A sin(000πmt), where 0 m 600 is the location on the broadcast dial and t is measured in seconds. For eample, a station at 980 on the AM dial has a function of the form f(t) = A sin(000π(980)t) = A sin(960000πt). Sound information is added to this signal by varying (modulating) A, that is, by changing the amplitude of the waves being transmitted. (AM means amplitude modulation. ) For a station at 980 on the dial, what is the period of the function f? What is the frequency (number of complete waves per second)?. (Question 48 from Section 6. of your tetbook) The number of hours of daylight in Winnipeg, Manitoba, can be approimated by d(t) = 4. sin(.0t.) +, where t is measured in days, with t = being January. (a) On what day is there the most daylight? The least? How much daylight is there on these days? (b) On which days are there hours or more of daylight? What do you think the period of this function is? Why? 4

10 0 Trigonometric Graphs Worksheet Concepts: Period, Amplitude, and Phase Shift The csc, sec, and cot Functions The Graphs of the csc, sec, and cot Functions (Sections 6. and 6.6). For each graph, (i) find the period if it is defined and (ii) find the amplitude if it is defined. (a) y (b) y (c) y

11 . For each of the following equations, what is the period, amplitude and phase shift of the graph? (a) y = 4 sin () ( ) (b) y = cos ( π (c) y = sin π ) ( (d) y = tan + π ) 4 (e) y = u cos(v + w). (Assume that u, v, and w are positive.). Evaluate the csc, sec, and cot functions at each of the following angles. (a) θ = π (b) θ = 9π 4 (c) θ = 4π (d) θ = π 6 4. (a) The terminal side of an angle, θ, in standard position contains the point (, 9). Evaluate the csc, sec, and cot functions at θ. (b) The terminal side of an angle, θ, in standard position contains the point (, 4). Evaluate the csc, sec, and cot functions at θ.. Suppose θ is in the fourth quadrant and cos(θ) =. Evaluate the csc, sec, and cot functions on θ. 6. Sketch the graphs of the following equations. Be sure that the graph is well-labeled. (a) y = csc () + ( ) (b) y = sec + ( (c) y = cot π ) 4. Use algebra and identities to simplify the epression. Assume all denominators are nonzero. (a) sin(t) tan(t) 8. Solve each of the following equations. (a) cos()=0 (b) sin t sint = 0 (c) cos t cost = (b) cos(t) sin(t)tan(t)

12 9. (Question 4 from Section 6. of your tetbook) Burke s blood pressure can be modeled by the function g(t) = cos(.πt) +, where t is the time (in seconds) and f(t) is in millimeters of mercury. The highest pressure (systolic) occurs when the heart beats, and the lowest pressure (diastolic) occurs when the heart is at rest between beats. The blood pressure is the ratio systolic/diastolic. (a) Graph the blood pressure function over a period of two seconds and determine Burke s blood pressure. (b) Find Burke s pulse rate (number of heartbeats per minute). (c) According to current guidelines, someone with systolic pressure above 40 or diastolic pressure above 90 has high blood pressure and should see a doctor about it. What would you advise the person in this case? 0. (Question 4 from Section 6. of your tetbook) The current generated by an AM radio transmitter is given by a function of the form f(t) = A sin(000πmt), where 0 m 600 is the location on the broadcast dial and t is measured in seconds. For eample, a station at 980 on the AM dial has a function of the form f(t) = A sin(000π(980)t) = A sin(960000πt). Sound information is added to this signal by varying (modulating) A, that is, by changing the amplitude of the waves being transmitted. (AM means amplitude modulation. ) For a station at 980 on the dial, what is the period of the function f? What is the frequency (number of complete waves per second)?. (Question 48 from Section 6. of your tetbook) The number of hours of daylight in Winnipeg, Manitoba, can be approimated by d(t) = 4. sin(.0t.) +, where t is measured in days, with t = being January. (a) On what day is there the most daylight? The least? How much daylight is there on these days? (b) On which days are there hours or more of daylight? What do you think the period of this function is? Why?

13 Trigonometric Identities Worksheet Concepts: Trigonometric Identities Reciprocal Identities Pythagorean Identities Periodicity Identities Negative Angle Identitites (Sections 6.6 &.). Find the period, phase shift, and sketch the graph of y = csc( π ).. Find all solutions to the equation tan(θ) = cot(θ) for 0 < θ < π. Use the Pythagorean identity for each of the following problems. (a) Find sin(t) if cos(t) = 0 and π < t < π. (b) Find cos(t) if sin(t) = and π < t < π. (c) Find cot(t) if sin(t) = and π < t < π. 4. Determine each of the following: (a) Find cos if sin = and tan > 0. (b) Find tan if cos = 4 and sin < 0. (c) Find tan if sec = 0 and sin > 0.

14 . Use the periodicity and negative angle identities to solve each of the following problems. (a) Find sin( t) if sin(t) =. (b) Find sin(t π) if sin(t) =. (c) Find cos( t + 4π) if cos(t) = 6. (d) Find cot(π t) if tan(t) =. 6. Use basic identities to simplify the epression. (a) cot θ sec θ sin θ (b) cos sin + csc sin (c) + sec cos cot (d) sin + tan + cos sin (e) sin csc cos (f) sin + sin cos (g) Prove the identity sin (tan cos cot cos ) = cos. State whether or not the equation might be an identity. If it appears to be, prove it. (a) cot = csc sec (b) sin( ) cos( ) = tan (c) + sec = tan (d) sin (cot + ) = cos (tan + ) (e) + sin sec + tan = sin sec tan (f) ( cos ) csc = sin (g) sec csc + sin cos = tan (h) csc = cos csc sin cos tan (i) = tan sin + cos

15 More Trigonometric Identities Worksheet Concepts: Trigonometric Identities Addition and Subtraction Identities Cofunction Identities Double-Angle Identities Power-Reducing Identitites Half-Angle Identities Product-Sum Identities (Sections. &.). Find the eact values of the following functions using the addition and subtraction formulas (a) sin 9π (b) cos π. Write the epression as the sine or cosine of an angle. (a) sin π cos π + cos π sin π (b) sin cos cos sin (c) cos cos sin sin. Simplify the following epressions as much as possible ( ) 9π (a) tan (b) sin( + y) sin( y) (c) cos( + y) cos( y) sin( + y) sin( y) (d) cos( + y) cos( y) ( (e) cos + π ) ( + sin π ) 6

16 4. Verify the following identity:. Verify the following identity: cos A cos B sin A + sin B + sin A sin B cos A + cos B = 0. cos( + y) cos( y) = cos sin y. 6. Use the cofunction identities to evaluate the following epression without using a calculator: sin ( ) + sin (6 ) + sin (69 ) + sin (9 ).. Find the values of the remaining trigonometric functions of if (a) sin = and the terminal point of is in Quadrant II. (b) tan = and cos > Simplify the epression sin 4 sin + sin. 9. Let f() = sin 4 + sin. Verify that f() = cos sin Use an appropriate half-angle formula to find the eact value of each epression. (a) sin π (b) cos π 8 (c) tan π (d) cos π 8 (e) sin π 8 cos 0. Use an appropriate half-angle formula to simplify.. Use an appropriate power-reducing formula to rewrite cos 4 sin in terms of the first power of cosine.. Write the product cos 6 cos as a sum. 4. Write the sum sin sin as a product.

17 Ma 0 Eam Review: Sections.4-.6, 6.-6., ,.-. Do not rely solely on this work sheet! Make sure to study homework problems, other work sheets, lecture notes, and the book!!!. Section.4 (suggested problems from HW - # s, 6,, 0) (a) Write as a single logarithm. log( ) + log( ) log(y ) (b) Simplify. e ln() ( ) z (c) Epress ln 4 in terms of ln(), ln(y) and ln(z). y. Section. (suggested problems from HW4 - # s,, 9, 0) (a) Solve for eactly. ln( + ) =. (b) Solve for eactly. e = (c) Section., question : Solve log( ) + log() = log(4) + log( + ) (d) How long until $0,000 doubles in a bank account with a yearly interest rate of r = % compounded continuously? (e) Section., question 69: The concentration of carbon dioide in the atmosphere is 64 parts per million (ppm) and is increasing eponentially at a continuous rate of.4%. How many years will it take for the concentration to reach 00 ppm?. Section.6 (suggested problems from HW - # s, 4, 6) (a) A culture starts with 8600 bacteria. After one hour the count is 0,000. i. Find a function that models the number of bacteria after t hours. ii. Find the number of bacteria after hours. iii. How long will it take for the number of bacteria to double? (b) The population of California was 0, 86, in 90 and, 668, 6 in 980. Assume the population grows eponentially. i. Find a function that models the population t years after 90. ii. Find the time required for the population to double. iii. In what year was the population, 000, 000? (c) The half-life of radium-6 is 600 years. Suppose we have a -mg sample. i. Find a function that models the mass remaining after t years. ii. How much of the sample will remain after 4000 years? iii. After how long will only 8 mg of the sample remain?

18 4. Section 6. (suggested problems from HW6 - # s,,, 6, 9) (a) Find the radian measure of a 40 angle. (b) What quadrant does the angle with measure 6π lie in. (c) Suppose ( that an angle of measure θ radians intersects the unit circle at the point ),. Give two possibilities for θ.. Section 6. (suggested problems from HW - # s,, 6) (a) Find the point P on the unit circle corresponding to the angle π 6. (b) Evaluate the sin, cos, and tan functions at t = π. (c) Evaluate sin, cos, and tan if the terminal ray of an angle contains the point (, ). (d) Approimate sin(.6). 6. Section 6.4 (suggested problems from HW8 - # s,, 6, ) (a) List the transformations ( needed to apply to the graph of y = sin() to sketch the ) graph of y = sin. Sketch the graph. Be sure that the graph is well-labeled. π (b) List the transformations ( needed to apply to the graph of y = cos() to sketch the graph of y = cos π ). Sketch the graph. Be sure that the graph is well-labeled. (c) The graph of a periodic function is shown below. Find a rule for the function. y π π π 4π

19 . Section 6. (suggested problems from HW9 - # s,, 4) (a) For each state the amplitude, period, phase shift and sketch the graph. i. f() = cos ( ) π ii. f() = tan ( ) + π 4 (b) How many solutions for between 0 and π does sin() = 0. have? 8. Section 6.6 (suggested problems from HW0 - # s,, 6) (a) Evaluate the si trigonometric functions at t = π. (b) Evaluate the si trigonometric functions if the terminal ray of an angle contains the point (.6,.8). (c) Answer as True or False. i. cos (t) = + sin (t). ii. cos(t π) = cos(t). iii. csc(t) = sin(t). iv. tan( t) = tan(t). (d) Find π < t < π such that sec(t) = eactly. (e) Find the other five trigonometric functions eactly if cos(t) = and π < t < π. 9. Section. (suggested problems from HW - # s 4, 8, 9, 0) (a) Simplify the following: ( ) i. tan() sin() + cot() cos() ( ) ii. cos(θ) sin(θ) iii. sec(t) sin(t) sin(t) cos(t) (b) Prove the following: i. csc () cos () csc () = ii. ( sec(θ) + )( sec(θ) ) = tan (θ) cos(α) iii. = sec(α) + tan(α) sin(α) sin(t) iv. cos(t) + cos(t) = csc(t) sin(t) v. sec 4 () tan 4 () = sec () + tan ()

20 0. Section. (suggested problems from HW - # s, 6, 8, 0) (a) Find the eact value of each of the following epressions: ( ) π i. cos ( ) 9π ii. sin ( ) π iii. tan (b) Write the following epression as a trigonometric function of one number, and find its eact value. ( ) ( ) ( ) ( ) π π π π cos cos + sin sin (c) Prove the cofunction identity: sin ( π θ) = cos(θ) (d) Section., Question 4: If is in the first and y is in the second quadrant, sin() = 4, and sin(y) = 4, find the eact value of sin( + y) and tan( + y) and the quadrant in which + y lies.. Section. (suggested problems from HW - # s,,, 4) (a) Find the eact value of each of the following epressions: ( ) π i. cos 8 ( ) π ii. sin 8 ( ) π iii. tan 8 (b) Given tan(t) = 4 and π < t < π, find sin(t), cos(t), and tan(t). (c) Given tan() = and is in Quadrant III, find sin ( ( ), cos ( ), and tan ). (d) Simplify. cos(4θ) sin(4θ) 4