Mathematics - Student Workbook Unit 8 = 7 6 Lesson : Degrees and Radians Approximate Completion Time: Days (cos, sin) Lesson : The Unit Circle Approximate Completion Time: Days y = asinb( - c) + d Lesson : Trigonometric Functions I Approximate Completion Time: Days h(t) Lesson : Trigonometric Functions II Approximate Completion Time: Days t UNIT FOUR Trigonometry I
Mathematics - Unit Student Workbook Complete this workbook by watching the videos on www.math.ca. Work neatly and use proper mathematical form in your notes. UNIT FOUR Trigonometry I
8 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example Define each term or phrase and draw a sample angle. Angle Definitions a) angle in standard position: Draw a standard position angle,. b) positive and negative angles: Draw = Draw = - c) reference angle: Find the reference angle of = 5. www.math.ca
Trigonometry LESSON ONE - Degrees and Radians 8 = 7 6 d) co-terminal angles: Draw the first positive co-terminal angle of 6. e) principal angle: Find the principal angle of =. f) general form of co-terminal angles: Find the first four positive co-terminal angles of = 5. Find the first four negative co-terminal angles of = 5. www.math.ca
8 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example Three Angle Types: Degrees, Radians, and Revolutions. a) Define degrees, radians, and revolutions. Angle Types and Conversion Multipliers i) Degrees: Draw = ii) Radians: Draw = rad iii) Revolutions: Draw = rev www.math.ca
Trigonometry LESSON ONE - Degrees and Radians 8 = 7 6 b) Use conversion multipliers to answer the questions and fill in the reference chart. Round all decimals to the nearest hundredth. Conversion Multiplier Reference Chart i) = rad degree radian revolution degree ii) = rev radian iii).6 = revolution iv).6 = rev v).75 rev = vi).75 rev = rad c) Contrast the decimal approximation of a radian with the exact value of a radian. i) Decimal Approximation: 5 = rad ii) Exact Value: 5 = rad www.math.ca
8 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example Convert each angle to the requested form. Round all decimals to the nearest hundredth. a) convert 75 to an approximate radian decimal. Angle Conversion Practice b) convert to an exact-value radian. c) convert to an exact-value revolution. d) convert.5 to degrees. e) convert to degrees. f) write as an approximate radian decimal. g) convert to an exact-value revolution. h) convert.5 rev to degrees. i) convert rev to radians. www.math.ca
Trigonometry LESSON ONE - Degrees and Radians 8 = 7 6 Example The diagram shows commonly used degrees. Find exact-value radians that correspond to each degree. When complete, memorize the diagram. Commonly Used Degrees and Radians a) Method One: Find all exact-value radians using a conversion multiplier. b) Method Two: Use a shortcut. (Counting Radians) 9 = = 5 = 5 = 6 = 5 = = = = 8 6 = = = 5 = = 5 = = = 7 www.math.ca
8 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example 5 a) Draw each of the following angles in standard position. State the reference angle. Reference Angles b) -6 c) 5. d) - 5 e) 7 www.math.ca
Trigonometry LESSON ONE - Degrees and Radians 8 = 7 6 Example 6 a) 9 Draw each of the following angles in standard position. State the principal and reference angles. Principal and Reference Angles b) -855 c) 9 d) - www.math.ca
8 = 7 6 Trigonometry LESSON ONE - Degrees and Radians For each angle, find all co-terminal Example 7 Co-terminal Angles angles within the stated domain. a) 6, Domain: -6 < 8 b) -95, Domain: -8 < 7 c).78, Domain: - < d) 8, Domain: < 7 5 www.math.ca
Trigonometry LESSON ONE - Degrees and Radians 8 = 7 6 Example 8 For each angle, use estimation to find the principal angle. a) 89 b) -7. Principal Angle of a Large Angle 9 c) d) 5 95 6 www.math.ca
8 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example 9 a) principal angle = (find co-terminal angle rotations counter-clockwise) Use the general form of co-terminal angles to find the specified angle. General Form of Co-terminal Angles b) principal angle = 5 (find co-terminal angle rotations clockwise) c) How many rotations are required to find the principal angle of -? State the principal angle. d) How many rotations are required to find the principal angle of? State the principal angle. www.math.ca
Trigonometry LESSON ONE - Degrees and Radians 8 = 7 6 Example Six Trigonometric Ratios In addition to the three primary trigonometric ratios (sin, cos, and tan), there are three reciprocal ratios (csc, sec, and cot). Given a triangle with side lengths of x and y, and a hypotenuse of length r, the six trigonometric ratios are as follows: sin = y r csc = sin = r y r y cos = x r sec = cos = r x x tan = y x cot = tan = x y a) If the point P(-5, ) exists on the terminal arm of an angle in standard position, determine the exact values of all six trigonometric ratios. State the reference angle and the standard position angle. b) If the point P(, -) exists on the terminal arm of an angle in standard position, determine the exact values of all six trigonometric ratios. State the reference angle and the standard position angle. www.math.ca
8 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example Determine the sign of each trigonometric ratio in each quadrant. Signs of Trigonometric Ratios a) sin b) cos c) tan 5 5 5 5 5 5 - - - - - - - 5 5 - - 5 5 - - 5 5 - d) csc e) sec f) cot 5 5 5 5 5 5 - - - - - - - 5 5 - - 5 5 - - 5 5 - g) How do the quadrant signs of the reciprocal trigonometric ratios (csc, sec, and cot) compare to the quadrant signs of the primary trigonometric ratios (sin, cos, and tan)? www.math.ca
Trigonometry LESSON ONE - Degrees and Radians 8 = 7 6 Example Given the following conditions, find the quadrant(s) where the angle could potentially exist. What Quadrant(s) is the Angle in? a) i) sin < ii) cos > iii) tan > b) i) sin > and cos > ii) sec > and tan < iii) csc < and cot > c) i) sin < and csc = ii) and csc < iii) sec > and tan = www.math.ca
8 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example Given one trigonometric ratio, find the exact values of the other five trigonometric ratios. State the reference angle and the standard position angle, to the nearest hundredth of a radian. Exact Values of Trigonometric Ratios a) b) www.math.ca
Trigonometry LESSON ONE - Degrees and Radians 8 = 7 6 Example Given one trigonometric ratio, find the exact Exact Values of values of the other five trigonometric ratios. Trigonometric Ratios State the reference angle and the standard position angle, to the nearest hundredth of a degree. a) b) www.math.ca
8 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example 5 Calculating with a calculator. Calculator Concerns a) When you solve a trigonometric equation in your calculator, the answer you get for can seem unexpected. Complete the following chart to learn how the calculator processes your attempt to solve for. If the angle could exist in either quadrant or... The calculator always picks quadrant I or II I or III I or IV II or III II or IV III or IV b) Given the point P(-, ), Mark tries to find the reference angle using a sine ratio, Jordan tries to find it using a cosine ratio, and Dylan tries to find it using a tangent ratio. Why does each person get a different result from their calculator? P(-, ) Mark s Calculation of (using sine) sin = 5 Jordan s Calculation of (using cosine) cos = - 5 Dylan s Calculation of (using tan) tan = - = 6.87 =. = -6.87 www.math.ca
Trigonometry LESSON ONE - Degrees and Radians 8 = 7 6 Example 6 Arc Length The formula for arc length is a = r, where a is the arc length, is the central angle in radians, and r is the radius of the circle. The radius and arc length must have the same units. r a) Derive the formula for arc length, a = r. a b) Solve for a, to the nearest hundredth. c) Solve for. (express your answer as a degree, to the nearest hundredth.) 6 cm 5 cm 5 cm a d) Solve for r, to the nearest hundredth. e) Solve for n. (express your answer as an exact-value radian.). cm 5 cm r 6 cm n www.math.ca
8 = 7 6 Trigonometry LESSON ONE - Degrees and Radians Example 7 Area of a circle sector. r a) Derive the formula for the area of a circle sector, A =. Sector Area r In parts (b - e), find the area of each shaded region. b) c) cm 7 6 cm d) e) 9 cm 6 6 cm cm www.math.ca
Trigonometry LESSON ONE - Degrees and Radians 8 = 7 6 Example 8 The formula for angular speed is, where ω (Greek: Omega) is the angular speed, is the change in angle, and T is the change in time. Calculate the requested quantity in each scenario. Round all decimals to the nearest hundredth. a) A bicycle wheel makes complete revolutions in minute. Calculate the angular speed in degrees per second. b) A Ferris wheel rotates in.5 minutes. Calculate the angular speed in radians per second. www.math.ca
8 = 7 6 Trigonometry LESSON ONE - Degrees and Radians c) The moon orbits Earth once every 7 days. Calculate the angular speed in revolutions per second. If the average distance from the Earth to the moon is 8 km, how far does the moon travel in one second? d) A cooling fan rotates with an angular speed of rpm. What is the speed in rps? e) A bike is ridden at a speed of km/h, and each wheel has a diameter of 68 cm. Calculate the angular speed of one of the bicycle wheels and express the answer using revolutions per second. www.math.ca
Trigonometry LESSON ONE - Degrees and Radians 8 = 7 6 Example 9 A satellite orbiting Earth km above the surface makes one complete revolution every 9 minutes. The radius of Earth is approximately 67 km. a) Calculate the angular speed of the satellite. Express your answer as an exact value, in radians/second. km 67 km b) How many kilometres does the satellite travel in one minute? Round your answer to the nearest hundredth of a kilometre. www.math.ca
(cos, sin) Trigonometry LESSON TWO - The Unit Circle Example Introduction to Circle Equations. a) A circle centered at the origin can be represented by the relation x + y = r, where r is the radius of the circle. Draw each circle: i) x + y = ii) x + y = 9 Equation of a Circle - - - - b) A circle centered at the origin with a radius of has the equation x + y =. This special circle is called the unit circle. Draw the unit circle and determine if each point exists on the circumference of the unit circle. i) (.6,.8) ii) (.5,.5) - - www.math.ca
Trigonometry LESSON TWO - The Unit Circle (cos, sin) c) Using the equation of the unit circle, x + y =, find the unknown coordinate of each point. Is there more than one unique answer? i) ii), quadrant II. iii) (-, y) iv), cos >. www.math.ca
(cos, sin) Trigonometry LESSON TWO - The Unit Circle Example The Unit Circle. The Unit Circle The following diagram is called the unit circle. Commonly used angles are shown as radians, and their exact-value coordinates are in brackets. Take a few moments to memorize this diagram. When you are done, use the blank unit circle on the next page to practice drawing the unit circle from memory. www.math.ca questions on next page.
Trigonometry LESSON TWO - The Unit Circle (cos, sin) a) What are some useful tips to memorize the unit circle? b) Draw the unit circle from memory using a partially completed template. www.math.ca
(cos, sin) Trigonometry LESSON TWO - The Unit Circle Example Use the unit circle to find the exact value of each trigonometric ratio. Finding Primary Trigonometric Ratios with the Unit Circle a) sin b) cos 8 c) cos 6 d) sin 6 e) sin f) cos g) sin h) cos - Example Use the unit circle to find the exact value of each trigonometric ratio. a) cos b) -cos c) sin 6 d) cos e) sin 5 9 f) -sin g) cos (-8 ) h) cos 7 www.math.ca
Trigonometry LESSON TWO - The Unit Circle (cos, sin) Example 5 Other Trigonometric Ratios. Other Trigonometric Ratios The unit circle contains values for cos and sin only. The other four trigonometric ratios can be obtained using the identities on the right. sec = cos csc = sin Given angles from the first quadrant of the unit circle, find the exact values of sec and csc. tan = sin cos cot = tan = cos sin a) sec sec = sec = sec = sec = 6 sec = b) csc csc = csc = csc = csc = 6 csc = www.math.ca
(cos, sin) Trigonometry LESSON TWO - The Unit Circle Example 6 Other Trigonometric Ratios. Other Trigonometric Ratios The unit circle contains values for cos and sin only. The other four trigonometric ratios can be obtained using the identities on the right. sec = cos csc = sin Given angles from the first quadrant of the unit circle, find the exact values of tan and cot. tan = sin cos cot = tan = cos sin a) tan tan = tan = tan = tan = 6 tan = b) cot cot = cot = cot = cot = 6 cot = www.math.ca
Trigonometry LESSON TWO - The Unit Circle (cos, sin) Example 7 a) sec sec Use symmetry to fill in quadrants II, III, and IV for each unit circle. = undefined sec = sec = sec = 6 b) csc Symmetry of the Unit Circle csc = csc = csc = csc = 6 sec = csc = undefined c) tan tan = undefined tan = tan = tan = 6 d) cot cot = cot = cot = cot = 6 tan = cot = undefined www.math.ca
(cos, sin) Trigonometry LESSON TWO - The Unit Circle Example 8 Find the exact value of each trigonometric ratio. Finding Reciprocal Trigonometric Ratios with the Unit Circle a) sec b) sec c) csc d) csc e) tan 6 5 f) -tan g) cot (7 ) h) cot 5 6 www.math.ca
Trigonometry LESSON TWO - The Unit Circle (cos, sin) Example 9 Find the exact value of each trigonometric expression. Evaluating Complex Expressions with the Unit Circle a) b) c) d) www.math.ca
(cos, sin) Trigonometry LESSON TWO - The Unit Circle Example Find the exact value of each trigonometric expression. Evaluating Complex Expressions with the Unit Circle a) b) c) d) www.math.ca
Trigonometry LESSON TWO - The Unit Circle (cos, sin) Example Find the exact value of each trigonometric ratio. Finding the Trigonometric Ratios of Large Angles with the Unit Circle a) b) c) d) www.math.ca
(cos, sin) Trigonometry LESSON TWO - The Unit Circle Example Verify each trigonometric statement with a calculator. Note: Every question in this example has already been seen earlier in the lesson. Evaluating Trigonometric Ratios with a Calculator a) b) c) d) e) f) g) h) www.math.ca
Trigonometry LESSON TWO - The Unit Circle (cos, sin) Example Answer each of the following questions related to the unit circle. Coordinate Relationships on the Unit Circle a) What is meant when you are asked to find on the unit circle? b) Find one positive and one negative angle such that P() = c) How does a half-rotation around the unit circle change the coordinates? If =, find the coordinates of the point halfway around the unit circle. 6 d) How does a quarter-rotation around the unit circle change the coordinates? If =, find the coordinates of the point a quarter-revolution (clockwise) around the unit circle. e) What are the coordinates of P()? Express coordinates to four decimal places. www.math.ca
(cos, sin) Trigonometry LESSON TWO - The Unit Circle Example Answer each of the following questions related to the unit circle. a) What is the circumference of the unit circle? Circumference and Arc Length of the Unit Circle b) How is the central angle of the unit circle related to its corresponding arc length? c) If a point on the terminal arm rotates from P() = (, ) to P() =, what is the arc length? d) What is the arc length from point A to point B on the unit circle? A B www.math.ca
Trigonometry LESSON TWO - The Unit Circle (cos, sin) Example 5 Answer each of the following questions related to the unit circle. a) Is sin = possible? Explain, using the unit circle as a reference. Domain and Range of the Unit Circle b) Which trigonometric ratios are restricted to a range of - y? Which trigonometric ratios exist outside that range? Range Number Line cos & sin csc & sec tan & cot www.math.ca
(cos, sin) Trigonometry LESSON TWO - The Unit Circle c) If exists on the unit circle, how can the unit circle be used to find cos? How many values for cos are possible? d) If exists on the unit circle, how can the equation of the unit circle be used to find sin? How many values for sin are possible? e) If cos =, and <, how many values for sin are possible? www.math.ca
Trigonometry LESSON TWO - The Unit Circle (cos, sin) Example 6 Complete the following questions related to the unit circle. Unit Circle Proofs a) Use the Pythagorean Theorem to prove that the equation of the unit circle is x + y =. b) Prove that the point where the terminal arm intersects the unit circle, P(), has coordinates of (cos, sin). c) If the point exists on the terminal arm of a unit circle, find the exact values of the six trigonometric ratios. State the reference angle and standard position angle to the nearest hundredth of a degree. www.math.ca
(cos, sin) Trigonometry LESSON TWO - The Unit Circle Example 7 In a video game, the graphic of a butterfly needs to be rotated. To make the butterfly graphic rotate, the programmer uses the equations: x = x cos - y sin y = x sin + y cos to transform each pixel of the graphic from its original coordinates, (x, y), to its new coordinates, (x, y ). Pixels may have positive or negative coordinates. a) If a particular pixel with coordinates of (5, ) is rotated by, what are the new 6 coordinates? Round coordinates to the nearest whole pixel. 5 b) If a particular pixel has the coordinates (6, 8) after a rotation of, what were the original coordinates? Round coordinates to the nearest whole pixel. www.math.ca
Trigonometry LESSON TWO - The Unit Circle (cos, sin) Example 8 From the observation deck of the Calgary Tower, an observer has to tilt their head A down to see point A, and B down to see point B. a) Show that the height of the observation x deck is h =. cot A - cot B h B A A B x b) If A =, B =, and x =.9 m, how high is the observation deck above the ground, to the nearest metre? www.math.ca
y = asinb( - c) + d Trigonometry LESSON THREE - Trigonometric Functions I a) Example Label all tick marks in the following grids and state the coordinates of each point. y Trigonometric Coordinate Grids 5-5 b) y - www.math.ca
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb( - c) + d c) y 8 8 - d) y - www.math.ca
y = asinb( - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example a) Draw y = sin. Exploring the graph of y = sin. y y = sin 5 7 5 5 7 6 6 6 6 7 5 5 7 5 6 6 6 6 - b) State the amplitude. c) State the period. d) State the horizontal displacement (phase shift). e) State the vertical displacement. Unit Circle Reference f) State the -intercepts. Write your answer using a general form expression. g) State the y-intercept. h) State the domain and range. www.math.ca
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb( - c) + d Example a) Draw y = cos. Exploring the graph of y = cos. y y = cos 5 7 5 5 7 6 6 6 6 7 5 5 7 5 6 6 6 6 - b) State the amplitude. c) State the period. d) State the horizontal displacement (phase shift). e) State the vertical displacement. Unit Circle Reference f) State the -intercepts. Write your answer using a general form expression. g) State the y-intercept. h) State the domain and range. www.math.ca
y = asinb( - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example a) Draw y = tan. Exploring the graph of y = tan. y y = tan 7 5 5 7 5 5 7 5 5 7 6 6 6 6 6 6 6 6 - - b) Is it correct to say a tangent graph has an amplitude? c) State the period. d) State the horizontal displacement (phase shift). e) State the vertical displacement. f) State the -intercepts. Write your answer using a general form expression. tan = - tan = - 5 tan = 6 - Unit Circle Reference tan = undefined tan = tan = tan = 6 g) State the y-intercept. h) State the domain and range. tan = 7 tan = 6 5 tan = tan = tan tan = tan = - 6 7 tan = - 5 tan = - = undefined www.math.ca
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb( - c) + d Example 5 Graph each function over the domain. The base graph is provided as a convenience. a) y = sin b) y = -cos The a Parameter 5 5-5 -5 c) y = sin 5 d) y = cos 5 5-5 -5 www.math.ca
y = asinb( - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Determine the trigonometric function corresponding to each graph. a) write a sine function. b) write a sine function. 8 Example 6 8 The a Parameter -8-8 c) write a cosine function. d) write a cosine function. 5 ( ), - -5 www.math.ca
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb( - c) + d 5 Example 7 a) y = sin - Graph each function over the domain. The base graph is provided as a convenience. b) y = cos + 5 The d Parameter -5-5 c) y = - sin + d) y = cos - 5 5-5 -5 www.math.ca
y = asinb( - c) + d Trigonometry LESSON THREE - Trigonometric Functions I a) write a sine function. Example 8 Determine the trigonometric function corresponding to each graph. b) write a cosine function. 5 The d Parameter - -5 c) write a cosine function. d) write a sine function. - - www.math.ca
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb( - c) + d Example 9 Graph each function over the stated domain. The base graph is provided as a convenience. a) y = cos ( ) b) y = sin ( ) The b Parameter - - c) y = cos ( 6) d) y = sin ( ) 5 6 6 8 - - www.math.ca
y = asinb( - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example Graph each function over the stated domain. The base graph is provided as a convenience. a) y = -sin() (- ) b) y = cos + 6 (- ) The b Parameter 6 - - - - -6 - - c) y = cos - (- ) d) y = sin ( 6) - - 6 - - - - www.math.ca
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb( - c) + d Example a) write a cosine function. Determine the trigonometric function corresponding to each graph. The b Parameter - b) write a cosine function. 6 - www.math.ca
y = asinb( - c) + d Trigonometry LESSON THREE - Trigonometric Functions I c) write a sine function. The b Parameter 6 9 - d) write a sine function. - www.math.ca
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb( - c) + d Example Graph each function over the stated domain. The base graph is provided as a convenience. The c Parameter a) (- ) b) (- ) - - - - - - c) (- ) d) (- ) - - - - - - www.math.ca
y = asinb( - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example Graph each function over the stated domain. The base graph is provided as a convenience. The c Parameter a) b) (- 6) - 6 - - c) (- ) d) (- ) - - - - - - www.math.ca
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb( - c) + d Example a) write a cosine function. Determine the trigonometric function corresponding to each graph. The c Parameter - b) write a sine function. 6-6 www.math.ca
y = asinb( - c) + d Trigonometry LESSON THREE - Trigonometric Functions I c) write a sine function. The c Parameter - - d) write a cosine function. -8-8 - www.math.ca
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb( - c) + d Example 5 Graph each function over the stated domain. The base graph is provided as a convenience. a, b, c, & d a) ( 6) b) ( ) 5 6-5 - c) - ( ) d) ( ) 5 6-5 -6 www.math.ca
y = asinb( - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example 6 Write a trigonometric function for each graph. a, b, c, & d a) - - - b) - - - www.math.ca
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb( - c) + d Example 7 a) Draw y = sec. Exploring the graph of y = sec. y Graphing Reciprocal Functions 5 7 5 5 7 6 6 6 6-7 5 5 7 5 6 6 6 6 - - b) State the period. c) State the domain and range. d) Write the general equation of the asymptotes. Unit Circle Reference (for sec) e) Given the graph of f() = cos, draw y =. f() y sec = - sec = - 5 sec = 6 - sec = undefined sec = sec = sec = 6 sec = - sec = - 7 sec = - 6 5 sec = - sec = - sec sec = 6 7 sec = 5 sec = = undefined www.math.ca
y = asinb( - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example 8 a) Draw y = csc. Exploring the graph of y = csc. y Graphing Reciprocal Functions 5 7 5 5 7 6 6 6 6-7 5 5 7 5 6 6 6 6 - - b) State the period. c) State the domain and range. d) Write the general equation of the asymptotes. Unit Circle Reference (for csc) e) Given the graph of f() = sin, draw y =. f() y csc = csc = 5 csc = 6 csc = csc = csc = csc = 6 csc = undefined csc = undefined - 7 csc = - 6 5 csc = - csc = - csc = - csc = - 6 7 csc = - 5 csc = - www.math.ca
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb( - c) + d Example 9 a) Draw y = cot. Exploring the graph of y = cot. y Graphing Reciprocal Functions 7 5 5 7 5 5 7 5 5 7 6 6 6 6 6 6 6 6 - - b) State the period. c) State the domain and range. d) Write the general equation of the asymptotes. Unit Circle Reference (for cot) e) Given the graph of f() = tan, draw y =. f() y cot = - cot = - 5 cot = 6 - cot = cot = cot = cot = 6 cot = undefined cot = undefined - 7 cot = 6 5 cot = cot = cot = cot = - 6 7 cot = - 5 cot = - www.math.ca
y = asinb( - c) + d Trigonometry LESSON THREE - Trigonometric Functions I Example Graph each function over the domain. The base graph is provided as a convenience. State the new domain and range. Transformations of Reciprocal Functions a) b) - y = sec - y = sec c) d) - y = csc - www.math.ca y = cot
Trigonometry LESSON THREE - Trigonometric Functions I y = asinb( - c) + d This page has been left blank for correct workbook printing. www.math.ca
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 Trigonometric Functions of Angles Trigonometric Functions of Angles a) b) i) Graph: ( < ) i) Graph: (º < 5º) y y 8º 6º 5º - y = cos (one cycle shown) - y = cos (one cycle shown) - - ii) Graph this function using technology. ii) Graph this function using technology. www.math.ca
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t a) h Example 7 i) Graph: Trigonometric Functions of Real Numbers. b) i) Graph: y Trigonometric Functions of Real Numbers 5 5 6 t 8 6 x - - - - ii) Graph this function using technology. ii) Graph this function using technology. c) What are three differences between trigonometric functions of angles and trigonometric functions of real numbers? www.math.ca
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 Determine the view window for each function and sketch each graph. Graph Preperation and View Windows a) b) www.math.ca
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t Example 7 Determine the view window for each function and sketch each graph. Graph Preperation and View Windows a) b) www.math.ca
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 5 a) write a cosine function. Determine the trigonometric function corresponding to each graph. Find the Trigonometric Function of a Graph 8 6 - b) write a sine function. 5-8 6-5 www.math.ca
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t c) write a cosine function. (8, 9) 5 (6, -) - d) write a sine function. (5, 5) (, -5) - www.math.ca
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 6 Answer the following questions: Assorted Questions a) If the transformation g() - = f() is applied to the graph of f() = sin, find the new range. b) Find the range of. c) If the range of y = cos + d is [-, k], determine the values of d and k. www.math.ca
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t d) State the range of f() - = msin() + n. e) The graphs of f() and g() intersect at the points and If the amplitude of each graph is quadrupled, determine the new points of intersection. www.math.ca
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 77 Answer the following questions: Assorted Questions a) If the point lies on the graph of, find the value of a. b) Find the y-intercept of. www.math.ca
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t c) The graphs of f() and g() intersect at the point (m, n). Find the value of f(m) + g(m). (m, n) g() n f() m d) The graph of f() = kcos is transformed to the graph of g() = bcos by a vertical stretch about the x-axis. If the point exists on the graph of g(), state the vertical stretch factor. k b f() g() www.math.ca
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 8 The graph shows the height of a pendulum bob as a function of time. One cycle of a pendulum consists of two swings - a right swing and a left swing. h(t) cm 8 cm cm ground level cm s s s s t a) Write a function that describes the height of the pendulum bob as a function of time. b) If the period of the pendulum is halved, how will this change the parameters in the function you wrote in part (a)? c) If the pendulum is lowered so its lowest point is cm above the ground, how will this change the parameters in the function you wrote in part (a)? www.math.ca
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t Example 7 9 A wind turbine has blades that are m long. An observer notes that one blade makes complete rotations (clockwise) every minute. The highest point of the blade during the rotation is 5 m. a) Using Point A as the starting point of the graph, draw the height of the blade over two rotations. A h(t) t b) Write a function that corresponds to the graph. c) Do we get a different graph if the wind turbine rotates counterclockwise? www.math.ca
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 A person is watching a helicopter ascend from a distance 5 m away from the takeoff point. a) Write a function, h(), that expresses the height as a function of the angle of elevation. Assume the height of the person is negligible. h 5 m b) Draw the graph, using an appropriate domain. h() c) Explain how the shape of the graph relates to the motion of the helicopter. www.math.ca
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t Example 7 A mass is attached to a spring m above the ground and allowed to oscillate from its equilibrium position. The lowest position of the mass is.8 m above the ground, and it takes s for one complete oscillation. a) Draw the graph for two full oscillations of the mass. h(t) t b) Write a sine function that gives the height of the mass above the ground as a function of time. c) Calculate the height of the mass after. seconds. Round your answer to the nearest hundredth. d) In one oscillation, how many seconds is the mass lower than. m? Round your answer to the nearest hundredth. www.math.ca
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 A Ferris wheel with a radius of 5 m rotates once every seconds. Riders board the Ferris wheel using a platform m above the ground. a) Draw the graph for two full rotations of the Ferris wheel. h(t) t b) Write a cosine function that gives the height of the rider as a function of time. c) Calculate the height of the rider after.6 rotations of the Ferris wheel. Round your answer to the nearest hundredth. d) In one rotation, how many seconds is the rider higher than 6 m? Round your answer to the nearest hundredth. www.math.ca
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t Example 7 The following table shows the number of daylight hours in Grande Prairie. December March June September December 6h, 6m h, 7m 7h, 9m h, 7m 6h, 6m a) Convert each date and time to a number that can be used for graphing. Day Number December = March = June = September = December = Daylight Hours 6h, 6m = h, 7m = 7h, 9m = h, 7m = h, 6m = b) Draw the graph for one complete cycle (winter solstice to winter solstice). d(n) n www.math.ca
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II c) Write a cosine function that relates the number of daylight hours, d, to the day number, n. d) How many daylight hours are there on May? Round your answer to the nearest hundredth. e) In one year, approximately how many days have more than 7 daylight hours? Round your answer to the nearest day. www.math.ca
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t Example 7 The highest tides in the world occur between New Brunswick and Nova Scotia, in the Bay of Fundy. Each day, there are two low tides and two high tides. The chart below contains tidal height data that was collected over a -hour period. Bay of Fundy Low Tide Time Decimal Hour Height of Water (m) : AM.8 Note: Actual tides at the Bay of Fundy are 6 hours and minutes apart due to daily changes in the position of the moon. High Tide 8: AM. In this example, we will use 6 hours for simplicity. Low Tide : PM.8 High Tide 8: PM. a) Convert each time to a decimal hour. b) Graph the height of the tide for one full cycle (low tide to low tide). h(t) t www.math.ca
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II c) Write a cosine function that relates the height of the water to the elapsed time. d) What is the height of the water at 6:9 AM? Round your answer to the nearest hundredth. e) For what percentage of the day is the height of the water greater than m? Round your answer to the nearest tenth. www.math.ca
Mice Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t Example 7 5 A wooded region has an ecosystem that supports both owls and mice. Owl and mice populations vary over time according to the equations: Owl population: Mouse population: where O is the population of owls, M is the population of mice, and t is the time in years. a) Graph the population of owls and mice over six years. Population 6 8 Owls 5 5 6 7 8 Time (years) b) Describe how the graph shows the relationship between owl and mouse populations. www.math.ca
h(t) t Trigonometry LESSON FOUR - Trigonometric Functions II Example 7 6 The angle of elevation between the 6: position and the : position of a historical building s clock, as measured from an observer standing on a hill, is. The observer also knows that he is standing m away from the clock, and his eyes are at the same height as the base of the clock. The radius of the clock is the same as the length of the minute hand. If the height of the minute hand s tip is measured relative to the bottom of the clock, what is the height of the tip at 5:8, to the nearest tenth of a metre? m www.math.ca
Trigonometry LESSON FOUR - Trigonometric Functions II h(t) t Example 7 Shane is on a Ferris wheel, and his height can be described by the equation. Tim, a baseball player, can throw a baseball with a speed of m/s. If Tim throws a ball directly upwards, the height can be determined by the equation h ball (t) = -.95t + t + If Tim throws the baseball 5 seconds after the ride begins, when are Shane and the ball at the same height? www.math.ca
Answer Key Example : Trigonometry Lesson One: Degrees and Radians a) The rotation angle between the initial arm and the terminal arm is called the standard position angle. b) An angle is positive if we rotate the terminal arm counterclockwise, and negative if rotated clockwise. c) The angle formed between the terminal arm and the x-axis is called the reference angle. 5 d) If the terminal arm is rotated by a multiple of 6 in either direction, it will return to its original position. These angles are called co-terminal angles. e) A principal angle is an angle that exists between and 6. 6 Note: For illustrative purposes, all diagram angles will be in degrees. f) The general form of co-terminal angles is c = p + n(6 ) using degrees, or c = p + n() using radians. 5, 5, 765, 5, 85 5, -5, -675, -5, -95 - Example : a) i. One degree is defined as /6 th of a full rotation. ii. One radian is the angle formed when the terminal arm swipes out an arc that has the same length as the terminal arm. One radian is approximately 57.. 57. iii. One revolution is defined as 6º, or pi. It is one complete rotation around a circle. Conversion Multiplier Reference Chart degree radian revolution rev degree 8 6 radian revolution 8 6 rev rev rev b) i.. rad ii..6 rev iii. 8.97 iv.. rev v. 7 vi..7 rad c) i..79 rad ii. / rad Example : a).5 rad b) 7/6 rad c) / rev d). e) 7 f).7 rad g) / rev h) 8 i) 6 rad Example : = = 5 = 5 9 = 6 = 5 = = Example 5: a) r = b) r = 8 c) r = 56 (or.98 rad) -6 d) r = 5 (or / rad) 5 5 e) r = 5 (or /7 rad) = 8 = 6 = 8 56 9 5 = = 5 = = 5 = = Example 6: a) p =, r = b) p = 5, r = 5 c) p = 56, r = d) p =, r = 6 (or p =.7, r =.) (or p = /, r = /) Example 7: = 7 a) = 6, = 6 p b) = -95, = 5 p c = -,, 78 c = -855, -5, 5, 585 6 56 6 5 5 Example 8: a) p = 9 b) p = 8 c) p = d) p = (or.58 rad) (or /5 rad) (or /6 rad) 9 5 8 c) = 675, = 5 p d) = 8, = p c = -5, 5 c = -96, -6, -,, (or c = -.785, 5.5) 8, 5 (or c = -6/, -/, -/, /, /, /) Example 9: a) c = 8 b) c = -8/5 c) c = d) c = / www.math.ca
Answer Key Example : a) p =.6, r = 67.8.6 b) p =.69, r = 56. -5 67.8 56. -.69 Example : a) sin: QI: +, QII: +, QIII: -, QIV: - b) cos: QI +, QII: -, QIII: -, QIV: + c) tan: QI +, QII: -, QIII: +, QIV: - d) csc: QI: +, QII: +, QIII: -, QIV: - e) sec: QI +, QII: -, QIII: -, QIV: + f) cot: QI +, QII: -, QIII: +, QIV: - g) sin & csc share the same quadrant signs. cos & sec share the same quadrant signs. tan & cot share the same quadrant signs Example : a) i. QIII or QIV ii. QI or QIV iii. QI or QIII b) i. QI ii. QIV iii. QIII c) i. none ii. QIII iii. QI Example : a) p =.6, r =.6 b) p = 5.6, r = 5.8 (or p =.5 rad, r =.9 rad) (or p =.7 rad, r =. rad) 5.6-5.6 -.6 7 5.8 Example : a) p =., r = 6.87 b) p = 6., r =.69 6.87 5 -.69 -. 6. Example 5: a) If the angle could exist in either quadrant or... I or II I or III I or IV II or III II or IV III or IV The calculator always picks quadrant I I I II IV IV b) Each answer is different because the calculator is unaware of which quadrant the triangle is in. The calculator assumes Mark s triangle is in QI, Jordan s triangle is in QII, and Dylan s triangle is in QIV. Example 6: a) The arc length can be found by multiplying the circumference by the sector percentage. This gives us: a = r / = r. b).5 cm c).59 d).6 cm e) n = 7/6 Example 7: a) The area of a sector can be found by multiplying the area of the full circle by the sector percentage to get the area of the sector. This gives us: a = r / = r /. b) 8/ cm c) cm d) 8/ cm e) 5 cm Example 8: a) 6 /s b).7 rad/s c). km d) 7 rev/s e).6 rev/s Example 9: a) /7 rad/s b) 68.5 km www.math.ca
Answer Key Trigonometry Lesson Two: The Unit Circle Example : a) i. ii. b) i. Yes ii. No (.6,.8) (.5,.5) c) i. ii. - - iii. y = iv. Example : See Video. Example : a) b) - c) d) e) f) g) h) Example 5: a) b) Example 6: a) b),,,, Example 7: See Video. Example : a) b) c) d) e) - f) g) h),,,,,,,,,,,, Example : a) C = b) The central angle and arc length of the unit circle are equal to each other. c) a = / d) a = 7/6 Example 5: a) The unit circle and the line y = do not intersect, so it's impossible for sin to equal. b) Range Number Line cos & sin csc & sec tan & cot c) d) 5.,.7 e) - - - y = Example 8: a) - b) undefined c) d) e) f) - g) h) Example 9: a) b) c) d) Example : a) b) c) d) Example : a) - b) c) undefined d) undefined Example 6: a) Inscribe a right triangle with side lengths of x, y, and a hypotenuse of into the unit circle. We use absolute values because technically, a triangle must have positive side lengths. Plug these side lengths into the Pythagorean Theorem to get x + y =. b) Use basic trigonometric ratios (SOHCAHTOA) to show that x = cos and y = sin. c) p = 67., r =.68 x y Example : See Video. Example : a) P(/) means "point coordinates at /". b) c) d) e) P() = (-.99,.) Example 7: a) (67, ) b) (-79, ) Example 8: a) See Video b) 6 m www.math.ca
Answer Key Example : Trigonometry Lesson Three: Trigonometric Functions I a) (-5/6, ), (-/6, -), (7/6, ) b) (-/, -), (/, 6), (7/, -8) c) (-6, 8), (-, -8), (, -) d) (-, ), (/, -), (5/, -) Example : a) y = sin b) a = c) P = d) c = e) d = f) = n, nεi g) (, ) h) Domain: ε R, Range: - y y Example : a) y = cos b) a = c) P = d) c = e) d = f) = / + n, nεi g) (, ) h) Domain: ε R, Range: - y y 5 7 5 5 7 6 6 6 6 7 5 5 7 5 6 6 6 6 5 7 5 5 7 6 6 6 6 7 5 5 7 5 6 6 6 6 - - Example : a) y = tan b) Tangent graphs do not have an amplitude. c) P = d) c = e) d = f) = n, nεi g) (, ) h) Domain: ε R, / + n, nεi, Range: y ε R y Example 5: a) b) 5 5 7 5 5 7 5 5 7 5 5 7 6 6 6 6 6 6 6 6 - - -5-5 Example 7: c) d) a) b) c) d) 5 5 5 5 5 5-5 -5-5 -5-5 -5 Example 6: Example 8: a) b) c) - d) 5 a) b) Example 9: a) b) c) d) c) d) 6 6 8 - - - - www.math.ca
Answer Key Example : Example : a) b) a) b) 6 - - - - - 6-6 - - - - c) d) c) d) - - 6 - - - - - - - - Example : a) b) Example : c) d) a) b) c) d) Example : a) b) Example 5: a) b) - - - - 5 - - c) d) 6-5 - c) d) - - - - 5 6 - - -5-6 Example 6: a) b) www.math.ca
Answer Key Example 7: a) y = sec b) P = c) Domain: ε R, / + n, nεi; Range: y -, y d) = / + n, nεi y y 5 7 5 5 7 6 6 6 6-7 5 5 7 5 6 6 6 6 - - - Example 8: a) y = csc b) P = c) Domain: ε R, n, nεi; Range: y -, y d) = n, nεi y y 5 7 5 5 7 6 6 6 6-7 5 5 7 5 6 6 6 6 - - - Example 9: a) y = cot b) P = c) Domain: ε R, n, nεi; Range: yεr d) = n, nεi y y 7 5 5 7 5 5 7 5 5 7 6 6 6 6 6 6 6 6 - - - Example : a) b) c) d) - - - - Domain: ε R, / + n, nεi; (or: ε R, / ± n, nεw) Range: y -/, y / Domain: ε R, / + n/, nεi; (or: ε R, / ± n/, nεw) Range: y -, y Domain: ε R, / + n, nεi; (or: ε R, / ± n, nεw) Range: y -, y Domain: ε R, n(), nεi; (or: ε R, ±n(), nεw) Range: y ε R www.math.ca
Answer Key Trigonometry Lesson Four: Trigonometric Functions II Example : a) b) Example : a) b) y y h y 8º 6º 5º 5 5 6 t 8 6 x - - - - - - - - Example : Example 5: a) b) a) - - -6-8 - - - -6-8 - - - -6 y 5 6 7 8 x -5-75 -5-75 -5 - -5 - -5 y 75 5 5 5 75 5 75 5 75 5 5 5 x b) c) d) Example 6: a) Example : a) b) b) c) d) e) 5 y 8 6 8 6 - - -6-8 5 6 7 8 9 x - - - y 8 6 8 6 5 x Example 7: a) b) c) d) Example 8: a) b) The b-parameter is doubled when the period is halved. The a, c, and d parameters remain the same. c) The d-parameter decreases by units, giving us d =. All other parameters remain unchanged. www.math.ca
Mice Answer Key Example 9: Example : a) h(t) 5 75 5 a) Decimal daylight hours: 6.77 h,.8 h, 7.8 h,.8 h, 6.77 h b) d(n) 6 = 5.55cos + +.95 65 c) d ( n) ( n ) d) 5.86 h e) 6 days 5 t 8 b) c) If the wind turbine rotates counterclockwise, we still get the same graph. Example : a) h() -5 5 5 5 5 n Example : a) Decimal hours past midnight:. h, 8. h,. h,. h b) h(t) 6 c) d).75 m e).% 8 9 b), Example : a) h(t) 5.. c) The angle of elevation increases quickly at first, but slows down as the helicopter reaches greater heights. The angle never actually reaches 9. Example 5: 8 6 t a) Population b) See Video. 6 M(t) Owls 8 O(t) 5.8 5 6 7 8 Time (years) b) c).86 m t d).6 s Example 6:.5 m h(t). (8,.5).5 Example : a) h(t) 5 5 6 t 6 Example 7: 5.6 s and 8. s h(t) 9 t 5 5 75 5 5 75 (5.6,.5) (8.,.) b) c) 8. m d) 6.78 s 5 5 6 t www.math.ca