Trigonometry - Weebly
Trigonometry210 180 LESSON ONE - Degrees and Radians6Lesson NotesExample 1: Define each term or phrase and draw a sample angle.a) Angle in standard position.b) Positive and negative angles. Drawc) Reference angle.d)Draw the first positive.Conversion Multiplier Reference Chart (Example 2)degreeradianrevolutiondegreeradiane) Principal angle.f)positiveFind the first fourrevolutionExample 2: Three Angle Types: Degrees, Radians, and Revolutions.a) i. Define degrees.ii. Define radians.iii. Define revolutions.b) Use conversion multipliers to answer the questions and fill in the reference chart.i.ii.iii.iv.v.vi.c) Contrast the decimal approximation of a radian with the exact value of a radian.i.(decimal approximation). ii.(exact value).Example 3: Convert each angle to the requested form. Round all decimals to the nearest hundredth.a)b)c)d)to degrees. e)to degrees.f)as an approximate radian decimal. g)h)to degrees. i)to radians.Example 4: The diagram shows commonly used degrees.When complete, memorize the diagram.a) Method One:a conversion multiplier.b) Method Two: Use a shortcut (counting radians).Example 5: Draw each of the following angles instandard position. State the reference angle.a)b)c)d)e)Example 6: Draw each of the following angles instandard position. State the principal andreference angles.a)b)c) 9 d)www.math30.ca
TrigonometryLESSON ONE - Degrees and Radians Lesson NotesExample 7:a)c),b)d)Example 8: For each angle, use estimation to find the principal angle.a)b)c)d)Example 9:a) pc) ccb)n and) d)pcp(Find(Find n andcp)Example 10: In addition to the threeprimary trigonometric ratios (sin , cos ,and tan ), there are three reciprocalratios (csc , sec , and cot ). Given atriangle with side lengths of x and y,and a hypotenuse of length r, the sixtrigonometric ratios are as a)in standard position, determine the exactvalues of all six trigonometric ratios. State the reference angle and the standard position angle.b)in standard position, determine the exactvalues of all six trigonometric ratios. State the reference angle and the standard position angle.Example 11: Determine the sign of each trigonometric ratio in each quadrant.a) sinb)c)d)e)f)g) How do the quadrant signs of the reciprocal trigonometric ratios (csc , sec , and cot ) compareto the quadrant signs of the primary trigonometric ratios (sin , cos , and tan )?Example 12: Given the following conditions, find the quadrant(s) where the anglea) i. sinii. cosb) i. sincosc) i. sincsccould potentially exist.iii. tan ii. secii. costaniii. cscand csciii. seccottanExample 13: 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.a)b)Example 14: 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 degree.a)b)www.math30.ca
TrigonometryLESSON ONE - Degrees and Radians Lesson NotesExample 15: Calculatingwith a calculator.If the angle could exist ineither quadrant or .a) When you solve a trigonometric equation inyour calculator, the answer you get for can seemunexpected. Complete the following chart to learnhow the calculator processes your attempt to solvefor .b)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 tangentratio. Why does each person get adifferent result from their calculator?The calculator alwayspicks quadrantI or III or IIII or IVII or IIIII or IVIII or IVMark’s Calculationofsin 35Jordan’s Calculationofcos -45Dylan’s Calculationoftan 3-4Example 16: The formula for arc length is a, 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.a)b)c)d)e)rnaExample 17: Area of a circle sector.a) Derive the formula for the area of a circle sector,b)c)r.Find the area of each shaded region.d)e)9 cmwww.math30.ca
TrigonometryLESSON ONE - Degrees and Radians Lesson NotesExample 18: The formula for angular speed is, whereCalculate the requested quantity in each scenario. Round all decimals to thenearest hundredth.a)Calculate the angular speed in degrees per second.b)c)in one second?d)e)angular speed of one of the bicycle wheels and express the answer using revolutions per second.Example 19:a) Calculate the angular speed of the satellite.b)www.math30.ca
TrigonometryLESSON TWO - The Unit Circle(cos , sin )Lesson NotesExample 1: Introduction to Circle Equations.10a) A circle centered at the origin can be represented by therelation x2 y2 r2, where r is the radius of the circle.Draw each circle: i. x2 y2 4 ii. x2 y2 49-101010b) A circle centered at the origin with a radius of 1 has theequation x2 y2 1. This special circle is called the unit circle.Draw the unit circle and determine if each point exists on thecircumference of the unit circle: i. (0.6, 0.8) ii. (0.5, 0.5)-10-1010-101c) Using the equation of the unit circle, x2 y2 1, find the unknowncoordinate of each point. Is there more than one unique answer?1-1-1i., quadrant II.ii.iii. (-1, y)iv., cos 0.Example 2: The following diagram is called the unit circle. Commonly used angles are shown asradians, and their exact-value coordinates are in brackets. Take a few moments to memorize thisdiagram. When you are done, use the blank unit circle on the next page to practice drawing theunit circle from memory.a) What are some useful tips to memorize the unit circle?b) Draw the unit circle from memory.Example 3: Use the unit circle to find the exactvalue of each expression.a) sinb) cos 180 c) cos6f) cose) sin 0g) sin2d) sin46h) cos -120 3Example 4: Use the unit circle to find the exactvalue of each expression.a) cos 420 b) -cos 3e) sin2f) -sin4c) sin6d) cosg) cos2 (-840 ) h) cos33www.math30.ca
TrigonometryLESSON TWO - The Unit Circle(cos , sin )Lesson NotesExample 5: The unit circle contains values for cos andsin only. The other four trigonometric ratios can beobtained using the identities on the right. Find the exactvalues of sec and csc in the first quadrant.Example 6: Find the exact values of tan and cotin the first quadrant.sec 1coscsc 1sintan sincoscot 1cos tansinExample 7: Use symmetry to fill in quadrants II, III,and IV for sec , csc tan and cot .Example 8: Find the exact value of each expression.a) sec 120 b) sec2c) csc3d) csce) tan4f) -tan6g) cot2(270 )4h) cotExample 9: Find the exact value of each expression.a)c)b)d)Example 10: Find the exact value of each expression.a)c)b)d)Example 11: Find the exact value of each expression.a)d)c)b)Example 12: Verify each trigonometric statement with a calculator.Note: Every question in this example has already been seen earlier in the lesson.a)e)b)c)f)d)g)www.math30.cah)26
TrigonometryLESSON TWO - The Unit Circle(cos , sin )Lesson NotesExample 13: Coordinate Relationships on the Unit Circlea) What is meant when you are asked to findon the unit circle?b) Find one positive and one negative angle such thatc) How does a half-rotation around the unit circle change the coordinates?6d) How does a quarter-rotation around the unit circle change the coordinates?3e) What are the coordinates of P(3)? Express coordinates to four decimal places.Example 14: Circumference and Arc Length of the Unit Circlea) What is the circumference of the unit circle?b) How is the central angle of the unit circle related toits corresponding arc length?c) If a point on the terminal arm rotates fromtoA(1, 0)Diagram forExample 14 (d)., what is the arc length?Bd) What is the arc length from point A to point Bon the unit circle?Example 15: Domain and Range of the Unit Circlea) Is sin 2 possible? Explain, using the unit circle as a reference.b) Which trigonometric ratios are restricted to a range ofWhich trigonometric ratios exist outside that range?c) If?exists on the unit circle, how can the unit circlebe used to find cos ? How many values for cos are possible?Chart for Example 15 (b).Rangecossincscsectand) Ifexists on the unit circle, how can the equation of theunit circle be used to find sin ? How many values for sin are possible?e) If cossin are possible?www.math30.caNumber Line
TrigonometryLESSON TWO - The Unit Circle(cos , sin )Lesson NotesExample 16: Unit Circle Proofsa) Use the Pythagorean Theorem to prove that the equation of the unit circle is x2 y2 1.b) Prove that the point where the terminal arm intersects the unit circle, P( ), hascoordinates of (cos , sin ).exists on the terminal arm of a unit circle, find the exact valuesc) If the pointof the six trigonometric ratios. State the reference angle and standard position angle to the nearesthundredth of a degree.Example 17: In a video game, the graphic of a butterfly needs to be rotated. To make thebutterfly graphic rotate, the programmer uses the equations:to transform each pixel of the graphic from its original coordinates, (x, y), to itsnew coordinates, (x’, y’). Pixels may have positive or negative coordinates.a) If a particular pixel with coordinates of (250, 100) is rotated by, what are the new6coordinates? Round coordinates to the nearest whole pixel.b) If a particular pixel has the coordinates (640, 480) after a rotation of, what were the4original coordinates? Round coordinates to the nearest whole pixel.Example 18: From the observation deck of the Calgary Tower, an observer has todown to see point B.ABa) Show that the height of the observationxdeck is h .cot A - cot BABb) IfA ,B , and x 212.92 m,how high is the observation deck above theground, to the nearest metre?hBAxwww.math30.ca
y asinb( - c) dTrigonometryLESSON THREE - Trigonometric Functions ILesson NotesExample 1: Label all tick marks in the following grids and state the coordinates of each point.a)b)yy20500-5-20c)d)yy124000-40-12Example 2: Exploring the graph of y sina) Draw y sinb) State the 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.g) State the y-intercept. h) State the domain and range.Example 3: Exploring the graph of y cosa) Draw y cosb) State the 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.g) State the y-intercept. h) State the domain and range.Example 4: Exploring the graph of y tana) Draw y tanb) 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.g) State the y-intercept. h) State the domain and range.www.math30.ca
Trigonometryy asinb( - c) dLESSON THREE - Trigonometric Functions ILesson NotesExample 5: The a Parameter. Graph each function over the domain 0a) y 3sinb) y -2cosc) y 1sin2d) y 2 .5cos2Example 6: The a Parameter. Determine the trigonometric function corresponding to each graph.a) write a sine function.b) write a sine function.8c) write a cosine function.28d) write a cosine function.15(0000-8-28-1-5Example 7: The d Parameter. Graph each function over the domain 0a) y sin - 2b) y cos 4c) y -1sin 22d) y 14)2 .11cos 22Example 8: The d Parameter. Determine the trigonometric function corresponding to each graph.a) write a sine function.b) write a cosine function.c) write a cosine function.d) write a sine function.4353240000-4-35-32-4Example 9: The b Parameter. Graph each function over the stated domain.a) y cos2c) y cosb) y sin313d) y sin15Example 10: The b Parameter. Graph each function over the stated domain.a) y -sin(3c) y 2cos12b) y 4cos2 6(-2-1(-2d) y sin(-243www.math30.ca
y asinb( - c) dTrigonometryLESSON THREE - Trigonometric Functions ILesson NotesExample 11: The b Parameter. Determine the trigonometric function corresponding to each graph.a) write a cosine function.b) write a sine function.c) write a cosine function.d) write a sine function.22410000-2-2-4-1Example 12: The c Parameter. Graph each function over the stated domain.a)(-4b)c)(-2d)(-4(-2Example 13: The c Parameter. Graph each function over the stated domain.a)22c)b)(-2d)(-(-2Example 14: The c Parameter. Determine the trigonometric function corresponding to each graph.a) write a cosine function.b) write a sine function.11c) write a sine function.d) write a cosine function.6402-1-1-6Example 15: a, b, c, & d Parameters. Graph each function over the stated domain.b)a)c)-3d)www.math30.ca-4
Trigonometryy asinb( - c) dLESSON THREE - Trigonometric Functions ILesson NotesExample 16: a, b, c, & d. Determine the trigonometric function corresponding to each graph.a) write a cosine function.b) write a cosine function.212-2-12yExample 17: Exploring the graph of y sec3a) Draw y secb) State the period. c) State the domain and range.d) Write the general equation of the asymptotes.-21e) Given the graph of f( ) cos-3yExample 18: Exploring the graph of y csc3a) Draw y cscb) State the period. c) State the domain and range.d) Write the general equation of the asymptotes.-2-3yExample 19: Exploring the graph of y cot3a) Draw y cotb) State the period. c) State the domain and range.d) Write the general equation of the asymptotes.-2-3Example 20: Graph each function over the domain 02 . State the new domain and range.c)b)d)33330000y sec-321e) Given the graph of f( ) tan-321e) Given the graph of f( ) sina)2y sec-3y cscwww.math30.ca-3y cot
Trigonometryh(t)LESSON FOUR - Trigonometric Functions IILesson NotestExample 1: Trigonometric Functions of Angles(0a) i. Graph: 3(0ºb) i. Graph:ii. Graph this function using technology. 540º)ii. Graph this function using technology.Example 2: Trigonometric Functions of Real Numbers.a) i. Graph:b) i. Graph:ii. Graph this function using technology.ii. Graph this function using technology.c) What are three differences between trigonometric functions of angles and trigonometricfunctions of real numbers?Example 3: Determine the view window for each function and sketch each graph.a)b)Example 4: Determine the view window for each function and sketch each graph.b)a)Example 5: Determine the trigonometric function corresponding to each graph.a) write a cosine function.b) write a sine function.10c) write a cosine function.105d) write a sine function.300(8, 9)(1425, 150)0816-48160250(16, -3)-10-52400(300, -50)-10-300Example 6: a) If the transformation g(b) Find the range of4.c) If the range of y 3cos d is [-4, k],determine the values of d and k.e) The graphs of f( ) and g( ) intersect at the pointsand.If the amplitude of each graph is quadrupled, determine the new points of intersection.www.math30.cad) State the range off( ) - 2 msin(2 ) n.
Trigonometryh(t)LESSON FOUR - Trigonometric Functions IILesson NotestExample 7: a) If the pointlies on the graph ofb) Find the y-intercept of, find the value of a.Graph for Example 7c.(m, n)g( )c) The graphs of f( ) and g( ) intersect atthe point (m, n). Find the value of f(m) g(m).nf( )d) The graph of f(mabout the x-axis.Graph for Example 7dkf( )g( )bIf the pointof g(exists on the graphstate the vertical stretch factor.22Example 8: The graph shows the height of a pendulum bobas a function of time. One cycle of a pendulum consists oftwo swings - a right swing and a left swing.a) Write a function that describes the height of the pendulumbob as a function of time.b) If the period of the pendulum is halved, how will thischange the parameters in the function you wrote in part (a)?c) If the pendulum is lowered so its lowest point is 2 cmabove the ground, how will this change the parameters inthe function you wrote in part (a)?h(t)Graph for Example 812 cm8 cm4 cm0 cmground level1s2s3s4sAExample 9: A wind turbine has blades that are 30 m long. An observer notesthat one blade makes 12 complete rotations (clockwise) every minute.The highest point of the blade during the rotation is 105 m.a) Using Point A as the starting point of the graph,draw the height of the blade over two rotations.b) Write a function that corresponds to the graph.c) Do we get a different graph if the wind turbine rotates counterclockwise?Example 10: A person is watching a helicopter ascendfrom a distance 150 m away from the takeoff point.a) Write a function, h( ), that expresses the height as a function of theangle of elevation. Assume the height of the person is negligible.b) Draw the graph, using an appropriate domain.c) Explain how the shape of the graph relates to the motion of the helicopter.www.math30.cah150 mt
Trigonometryh(t)LESSON FOUR - Trigonometric Functions IILesson NotestExample 11: A mass is attached to a spring 4 m above the ground and allowed to oscillate from itsequilibrium position. The lowest position of the mass is 2.8 m above the ground, and it takes 1 sfor one complete oscillation.a) Draw the graph for two full oscillations of the mass.b) Write a sine function that gives the height of the massabove the ground as a function of time.c) Calculate the height of the mass after 1.2 seconds.Round your answer to the nearest hundredth.d) In one oscillation, how many seconds is the mass lowerthan 3.2 m? Round your answer to the nearest hundredth.Example 12: A Ferris wheel with a radius of 15 m rotates once every 100 seconds.Riders board the Ferris wheel using a platform 1 m above the ground.a) Draw the graph for two full rotations of the Ferris wheel.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 1.6 rotations of the Ferris wheel.Round your answer to the nearest hundredth.d) In one rotation, how many seconds is the rider higher than 26 m?Round your answer to the nearest hundredth.Example 13: The following table shows the number of daylight hours in Grande Prairie.December 216h, 46mMarch 2112h, 17mJune 2117h, 49mSeptember 21 December 2112h, 17m6h, 46ma) Convert each date and time to a number that can be used for graphing.Day Number December 21 March 21 June 21 September 21 December 21 Daylight Hours 6h, 46m 12h, 17m 17h, 49m 12h, 17m 12h, 46m b) Draw the graph for one complete cycle (winter solstice to winter solstice).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 2? Round your answer to the nearest hundredth.e) In one year, approximately how many days have more than 17 daylight hours?Round your answer to the nearest day.www.math30.ca
Trigonometryh(t)LESSON FOUR - Trigonometric Functions IILesson NotestExample 14: The highest tides in the world occur between New Brunswick andNova Scotia, in the Bay of Fundy. Each day, there are two low tides andtwo high tides. The chart below contains tidal height data that was collectedover a 24-hour period.TimeDecimal HourHeight of Water (m)3.48a) Convert each time to a decimal hour. Low Tide 2:12 AMHigh Tide 8:12 AM13.32b) Graph the height of the tide for oneLow Tide 2:12 PM3.48full cycle (low tide to low tide).13.32High Tide 8:12 PMc) Write a cosine function that relatesthe height of the water to the elapsed time.d) What is the height of the water at 6:09 AM? Round your answer to thenearest hundredth.e) For what percentage of the day is the height of the water greater than 11 m?Round your answer to the nearest tenth.Bay ofFundyBay ofFundyNote: Actual tides atthe Bay of Fundy are6 hours and 13 minutesapart due to dailychanges in the positionof the moon.In this example, we willuse 6 hours for simplicity.Example 15: 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.b) Describe how the graph shows the relationship between owl and mouse populations.Example 16: The angle of elevation between the 6:00 position and the 12:00 position.of a historical building’s clock, as measured from an observer standing on a hill, is444The observer also knows that he is standing 424 m away from theclock, 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 thebottom of the clock, what is the height of the tip at 5:08,to the nearest tenth of a metre?Example 17: Shane is on a Ferris wheel, and his heightcan be described by the equation.Tim, a baseball player, can throw a baseball with a speed of 20 m/s.If Tim throws a ball directly upwards, the height can be determinedby the equation hball(t) -4.905t2 20t 1.If Tim throws the baseball 15 seconds after the ride begins,when are Shane and the ball at the same height?www.math30.ca444424m
TrigonometryLESSON FIVE - Trigonometric EquationsLesson NotesExample 1: Primary Ratios. Find all angles in the domain 02 that satisfy the given equation.Write the general solution. Solve equations non-graphically using the unit circle.a)b)0c)d) tan2Example 2: Primary Ratios. Find all angles in the domain 02 that satisfy the given equation.Write the general solution. Solve equations graphically with intersection points.c)b)a)f)e)d)undefinedExample 3: Primary Ratios. Find all angles in the domain 0 360 that satisfy the given equation.Write the general solution. Solve equations non-graphically with a calculator (degree mode).a)c)b)Example 4: Primary Ratios. Find all angles in the domain 0Solve equations graphically with -intercepts.2 that satisfy the given equation.b)a)-Example 5: Primary Ratios. Solvea)cos feature of a calculator.02b)using the unit circle.c)d)usingb)using the unit circle.c)d)usingExample 6: Primary Ratios.a)cos feature of a calculator.Example 7: Reciprocal Ratios. Find all angles in the domain 02 that satisfy the given equation.Write the general solution. Solve equations non-graphically using the unit circle.a)b)c)Example 8: Reciprocal Ratios. Find all angles in the domain 02 that satisfy the given equation.Write the general solution. Solve equations graphically with intersection points.a)d)c)b)e)f)Example 9: Reciprocal Ratios. Find all angles in the domain 0 360 that satisfy thegiven equation. Write the general solution. Solve non-graphically with a calculator (degree mode).a)c)b)Example 10: Reciprocal Ratios. Find all angles in the domain 02 that satisfy thegiven equation. Write the general solution. Solve equations graphically with -intercepts.a)θb)θwww.math30.ca
TrigonometryLESSON FIVE - Trigonometric EquationsLesson NotesExample 11: Reciprocal Ratios. Solve csc0b)using the unit circle.a)cos feature of a calculator.2c)Example 12: Reciprocal Ratios. Solve seca)cos feature of a calculator.d)using0 b)using the unit circle.360 c)d)usingExample 13: First-Degree Trigonometric Equations. Find all angles in the domain 0satisfy the given equation. Write the general solution.a)c)b)d)Example 14: First-Degree Trigonometric Equations. Find all angles in the domain 0satisfy the given equation. Write the general solution.b)a)c)b) 4cos2c) 2cos2d) tan4c) 2sin3b) csc22 that2Example 16: Second-Degree Trigonometric Equations. Find all angles in the domain 0satisfy the given equation. Write the general solution.a) 2sin22 thatd)Example 15: Second-Degree Trigonometric Equations. Find all angles in the domain 0satisfy the given equation. Write the general solution.a) sin22 that2 that2Example 17: Double and Triple Angles.a)020b)2Example 18: Half and Quarter Angles.a)040b)8Example 19:a)b)Example 20:dda).b)c)of rotation (in degrees)?www.math30.ca
TrigonometryLESSON SIX - Trigonometric Identities ILesson NotesExample 1: Understanding Trigonometric Identities.a) Why are trigonometric identities considered to be a special type of trigonometric equation?b) Which of the following trigonometric equations are also trigonometric identities?i.ii.iii.iv.v.Example 2: The Pythagorean Identities.a) Using the definition of the unit circle, derive the identity sin2x cos2x 1.Why is sin2x cos2x 1 called a Pythagorean Identity?b) Verify that sin2x cos2x 1 is an identity using (i) x and (ii) x .c) Verify that sin2x cos2x 1 is an identity using a graphing calculator to draw the graph.d) Using the identity sin2x cos2x 1, derive 1 cot2x csc2x and tan2x 1 sec2x.e) Verify that 1 cot2x csc2x and tan2x 1 sec2x are identities for x .f) Verify that 1 cot2x csc2x and tan2x 1 sec2x are identities graphically.Example 3: Reciprocal Identities. Prove that each trigonometric statement is an identity.State the non-permissible values of x so the identity is true.a)b)Example 4: Reciprocal Identities. Prove that each trigonometric statement is an identity.State the non-permissible values of x so the identity is true.a)b)Example 5: Pythagorean Identities. Prove that each trigonometric statement is an identity.State the non-permissible values of x so the identity is true.a)b)d)c)Example 6: Pythagorean Identities. Prove that each trigonometric statement is an identity.State the non-permissible values of x so the identity is true.b)a)c)d)www.math30.ca
TrigonometryLESSON SIX- Trigonometric Identities ILesson NotesExample 7: Common Denominator Proofs. Prove that each trigonometric statement is an identity.State the non-permissible values of x so the identity is true.a)b)c)d)Example 8: Common Denominator Proofs. Prove that each trigonometric statement is an identity.State the non-permissible values of x so the identity is true.a)b)c)d)Example 9: Assorted Proofs. Prove each identity. For simplicity, ignore NPV’s and graphs.a)b)c)d)Example 10: Assorted Proofs. Prove each identity. For simplicity, ignore NPV’s and graphs.a)b)c)d)Example 11: Assorted Proofs. Prove each identity. For simplicity, ignore NPV’s and graphs.a)b)c)d)Example 12: Exploring the proof ofa) Prove algebraically thatb) Verify that.for3.c) State the non-permissible values ford) Show graphically that. Are the graphs exactly the same?www.math30.ca
TrigonometryLESSON SIX - Trigonometric Identities ILesson NotesExample 13: Exploring the proof of.a) Prove algebraically that.forb) Verify that3.c) State the non-permissible values for.d) Show graphically that. Are the graphs exactly the same?Example 14: Exploring the proof ofa) Prove algebraically thatforb) Verify that2.c) State the the non-permissible values for. Are the graphs exactly the same?d) Show graphically thatExample 15: Equations with Identites.a)c)b)Example 16: Equations with Identites.a)b)c)d)Example 17: Equations with Identites.a)b)c)d)www.math30.cad)
TrigonometryLESSON SIX- Trigonometric Identities ILesson NotesExample 18: Use the Pythagorean identities to find the indicated value anddraw the corresponding triangle.a) If the value offind the value of cosx within the same domain.b) If the value ofc)cos , find the value of secA within the same domain.7, and cot 0,7Example 19: Trigonometric Substitution.a) Using the triangle to the right, show thatcan be expressed as3.Hint: Use the triangle to find a trigonometric expression equivalent to b.bab) Using the triangle to the right, show thatcan be expressed as4.Hint: Use the triangle to find a trigonometric expression equivalent to a.www.math30.caa
TrigonometryLESSON SEVEN - Trigonometric Identities IILesson NotesExample 1: Evaluate each trigonometric sum or difference.c)b)a)e)d)f)Example 2: Write each expression as a single trigonometric ratio.a)c)b)Example 3: Find the exact value of each expression.a)b)d) Given the exact values of cosine and sine for 15 ,fill in the blanks for the other angles.c)Example 4: Find the exact valueof each expression.a)b)Example 3dc)Example 5: Double-angle identities.a) Prove the double-angle sine identity, sin2x 2sinxcosx.b) Prove the double-angle cosine identity, cos2x cos2x - sin2x.c) The double-angle cosine identity, cos2x cos2x - sin2x, can beexpressed as cos2x 1 - 2sin2x or cos2x 2cos2x - 1. Derive each identity.d) Derive the double-angle tan identity,.Example 6: Double-angle identities.a) Evaluate each of the following expressions using a double-angle identity.i.ii.iii.b) Express each of the following expressions using a double-angle identity.i.ii.iii.iv.c) Write each of the following expression as a single trigonometric ratio using a double-angle identity.i.ii.iii.www.math30.caiv.
TrigonometryLESSON SEVEN- Trigonometric Identities IILesson NotesNote: Variable restrictions may beignored for the proofs in this lesson.Example 7: Prove each trigonometric identity.a)b)c)d)Example 8: Prove each trigonometric identity.a)b)c)d)Example 9: Prove each trigonometric identity.a)b)c)d)Example 10: Prove each trigonometric identity.a)b)c)d)Example 11: Prove each trigonometric identity.a)b)d)c)Example 12: Prove each trigonometric identity.b)a)c)d)www.math30.ca
TrigonometryLESSON SEVEN - Trigonometric Identities IILesson NotesExample 13: Prove each trigonometric identity.a)b)d)c)Example 14:a)b)d)c)Example 15:a)b)Diagram forExample 18d)c)AExample 16:a)b)c)d)BExample 17:a)c)Cb)d)Diagram forExample 19Example 18: Trigonometric identities and geometry.xa) Show thatb) If A 32 and B 89 ,what is the value of C?57176Example 19: Trigonometric identities and geometry.104Solve for x. Round your answer to the nearest tenth.BA153www.math30.ca
www.math30.ca Trigonometry LESSON TWO - The Unit Circle Lesson Notes a) A circle centered at the origin can be represented by the relation x2 y2 r2, where r is the radius of the circle. Draw each circle: i. x2 y2 4 ii. x2 y2 49 Example 1: Introduction to Circle Equations. (cos f, sin f)-10-10 10-10 10-10 10 10 b) A circle centered at the origin with a radius
CHAPTER 1 1 Angles and Applications 1.1 Introduction Trigonometry is the branch of mathematics concerned with the measurement of the parts, sides, and angles of a triangle. Plane trigonometry, which is the topic of this book, is restricted to triangles lying in a plane. Trigonometry is based on certain ratio
1 Right Triangle Trigonometry Trigonometry is the study of the relations between the sides and angles of triangles. The word “trigonometry” is derived from the Greek words trigono (τρ ιγων o), meaning “triangle”, and metro (µǫτρω ), meaning “measure”. Though the ancient Greeks, such as Hipparchus
1 Review Trigonometry – Solving for Sides Review Gr. 10 2 Review Trigonometry – Solving for Angles Review Gr. 10 3 Trigonometry in the Real World C2.1 4 Sine Law C2.2 5 Cosine Law C2.3 6 Choosing between Sine and Cosine Law C2.3 7 Real World Problems C2.4 8 More Real World Problems C2.4 9
Trigonometry Standard 12.0 Students use trigonometry to determine unknown sides or angles in right triangles. Trigonometry Standard 5.0 Students know the definitions of the tangent and cotangent functions and can graph them. Trigonometry Standard 6.0 Students know the definitions of the secant and cosecant functions and can graph them.
Key words: GeoGebra, concept teaching, trigonometry, periodicity. INTRODUCTION Trigonometry teaching Being one of the major subjects in high school mathematics curriculum, trigonometry forms the basis of many advanced mathematics courses. The knowledge of trigonometry can be also used in physics, architecture and engineering.
Isosceles Right Triangles 14 30 -60 -90 15 Mixed practice 16-17 Trigonometry 18 Trigonometry 21 Holiday 22 Trigonometry 23-24 REVIEW Begin Test 25 TEST Tuesday, 1/8 Pythagorean Theorem 1. I can solve for the missing hypotenuse of a right triangle. 2. I can solve for the missing leg of a right triangle. 3. I can identify Pythagorean Triples.
SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles. Day 1 Basic Trigonometry Review Warm Up: Review the basic Trig Rules below and complete the example below: Basic Trigonometry Rules: These formulas ONLY work in a right triangle.
TIPS4RM: Grade 10 Applied: Unit 2 – Trigonometry (August 2008) 2-1 Unit 2 Grade 10 Applied. Trigonometry . Lesson Outline . . Introduce the inverse trigonometric ratio key on a scientific calculator. Students check how close their answers are from the Day 2 Home Activity.
