Formal & Informal Geometry
Formal & Informal Geometry MATH 3305
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Math 3305 Trigonometry Intro du ction Similar Right Triangles De nition and AA Similarity Theorem Discovery Lesson 7 Ratios of Sides of Similar Triangles Sine Cosine and Tangent of an Angle De nitions Isosceles right triangles 30 7 60 90 triangles Chart Summary Arbitrary right triangles Using the arc function key Law of Sines and Law of Cosines Law of Sines Discovery Lesson 7 Using the Law of Sines Discovery Lesson 7 The Ambiguity of the Arcsine Function with Arbitrary Triangles Law of Cosines Area of a Triangle Unit Circle Trigonometry Pythagorean Identity The Coordinate Connection Quadrantal Angles Referencing Angles 30 45 60 Graphing the Sine Function Graphing the Tangent Function Answers to Exercises F 06 updated Intro du ction Trigonometry is an ancient area of study Much of trigonometry was discovered and well known before the birth of Christ A direct translation of the word trigonometry yields the information that we are working with the measurement of 3 angles Indeed we will do that and a little bit more Initially we will work with right triangles and ratios of the side measures Later we ll expand to arbitrary triangles and nally we ll survey the functional approach to trigonometry This module is intended as both an overview and a review It is anticipated that most students will have encountered trigonometry before Some familiarity with a calculator that has both trig functions and inverse trig functions is posited Similar Right Triangles De nition of Similar Triangles Given two triangles AA and AB we say AA is similar to AB denoted AA 7 AB if and only if corresponding angles are congruent and the measures of corresponding sides are in the same proportion The Euclidean Geometry AA Similarity Theorem states that if two triangles have two pairs of corresponding congruent angles then the triangles are similar vnet 4 right triangles Here are four right triangles 7 with angles A B C and D being 90 7 which are similar C B D A A Aquot B Bquot c Cquot D Dquot NA 100cm 33 200 cm CC 300 cm D39D 200 cm AA 133 cm 33 267 cm coquot 400 cm DDquot 200 cm AW 167 cm 33 333 cm CCquot 500 cm D39Dquot 283 cm mzA39AquotA 3687 szBquotB39 3687 14000 3637 sz39D39D 4500 The similar triangles are part ofa famous set the 3 7 4 7 5 triangles The constant of proportionality k is the multiplier used to go from one triangle s side lengths to a similar triangle s side lengths What are the constants of proportionality among the similar triangles What can be said about the dissimilar one vnet nested triangles with a parallel side Here is an arrangement of two or more triangles that lets you see that they are similar right away If you know that AB and DE are parallel then you know that AABC and ADCE are similar Why is this true Similar Triangles exercise Given AA N AB What does this statement guarantee What is the constant of proportionality from AA to AB from AB to AA Triangle A Triangle B AAquot 250 cm 39Aquot39 103 cm Discovery Lesson Ratios of sides of Similar Right Triangles In Sketchpad create two similar triangles and measure one corresponding angle and the sides of each Create ratios of sides with one chosen angle as the angle of interest Compare these ratios to discover sine cosine and tangent of the chosen angle Using the following directions create a scalene right triangle reminder scalene means all three sides are different lengths Put a point on the sketch with the Point Tool and leave it selected Go to the Transformation menu and pick Translate Translate the point 2 cm in the fixed angle 0 direction Once this is done deselect the point and reselect your original point Translate it 35 cm in the 90 direction Label your points by selecting them in order A right angle B to the right and C above A while holding down the Shift key then go to the display menu and select label poin Then select each point individual and double click on it 7 change the name in the dialog box Select A B and C simultaneously by holding the Shift key and clicking on the points in turn then release the Shift key and go to the Construct menu and select segments Now you should have scalene AABC with the points labeled Construction notes and reminders To construct a triangle similar to AABC select the vertices A B and C and translate them 6 cm horizontally fixed angle 0 Double click on the translated point A and single click on the translations of B and C then go to the Transformation menu and select Dilate Insert the ratio 2 over 1 into the appropriate places in the dialog box and nish the transformation Deselect everything by clicking on the white part of the sketch Hide the two points used initially in the dilation by selecting them and going to the Display menu and picking hide points Do not delete them if you do you ll delete everything from the initial transformation onward Connect the vertices of the larger triangle using the Construct menu as before Label your corresponding points with primes A B and C C Construction notes and reminders Measure the sides by selecting the pair of vertices that serve and segment endpoints and going to the Measure menu select distance Do this for all siX sides What is the constant of proportionality from the initial triangle to the nal triangle Use the text tool to note that on the sketch Measure angle B and angle B by selecting vertices C B A in this order and going to the Measurement menu to measure angle Do the same for the primed sides C C A B A39 B39 C39A39 700 cm CA 350 cm A39B39 400 cm AB 200 cm C39B39 806 cm CB 403 cm mzC39B39A39 60260 mzCBA 60260 The constant of proportionalityfrom the original triangle to the final triangle is 2 Construction notes and reminders Now we ll get down to the trigonometry part Our angle of interest for the exercise will be the angle that measures 6026 We will nd out 3 facts about this angle for each of the two similar triangles The sine of angle B denoted sinB or sin B is the ratio W hypotenuse 39d d39 t B The cos1ne of angle B denoted cosB or cos B 1s the ratio w hypotenuse 39d 39t B The tangent of angle B denoted tanB or tan B 1s the ratio w s1de ad acent B Calculate these using the measurements on the Discovery Lesson sketch and under Measurement the calculator in Sketchpad sin B sin B cos B cos B tan B tan B Why are the ratios the same Sine Cosine and Tangent of an Angle In the Discovery Lesson we found that certain ratios of similar right triangles are dependably the same in similar triangles This gives rise to the universality of trigonometry If you are building a bridge or a pyramid you can make a model that ts on a tabletop and then using trigonometry and similarity build the real thing checking as you go that the ratios are the same De nitions If we have right triangle ABC and A is the right angle then 39d 39t B o the sme of angle B denoted s1nB or sm B is the ratio w hypotenuse 39d d39 t B o the cos1ne of angle B denoted cosB or cos B is the ratio w hypotenuse 39d 39t B o the tangent of angle B denoted tanB or tan B is the ratio w s1de adjacent B vnet the algebra of tangent Note that tangent is sine divided by cosine 7 check out the algebra of it Note too that there are three more trigonometric ratios that we will not use much in this class There is a table to be lled in later in the text Knowing that they exist and can be calculated is generally enough for a survey course cosecant B 7 the reciprocal of sin B secant B 7 the reciprocal of cos B cotangent B 7 the reciprocal of tan B We will explore three types of right triangles in this section 0 isosceles right triangles o 30 7 60 7 90 triangles o arbitrary right triangles vnet isosceles right triangles Isosceles right triangles These have a nice feature they are all similar to one another Why is this true Now let s look at the basic and nicest isosceles right triangle with sides oflength 1 and hypotenuse a different length B I A C BA 100 cm Note the calculator approximation for the AC 2100 cm hypotenuse Ifwe use the Pythagorean 7 BC 141 cm Theorem whatdo we nd forthe hypotenuse szCA 45000 What is the measurement of angle B How do you know How about angle C How do you know Let s calculate the hypotenuse using the Pythagorean Theorem First state the Pythagorean Theorem Next use it In this section we ll note that Sketchpad uses approximations of irrational numbers but we re going to use the actual irrational number and symbol for it Recall that 39d 39t B the s1ne of angle B denoted s1nB or s1n B 1s the ratio w hypotenuse 39d d39 t B the cos1ne of angle B denoted cosB or cos B 1s the ratio w hypotenuse 39d 39t B the tangent of angle B denoted tanB or tan B 1s the ratio w s1de adjacent B sin B cos B tan B Are these numbers the same for angle C Why or why not Now let s look at another isosceles right triangle M What are the measurements of angles H and Q What is sin Q What is cos H H What is tan H MH 400 cm MQ 400 cm mzHMQ 9000O Why are these the same as in the rst example Summarizing what we ve discovered sin 45 cos 45 tan 45 Isosceles Right triangle exercise What is the exact length of the hypotenuse in isosceles right triangle ABC What are the exact values of the sine cosine and tangent for A A A A B C AB 300 cm Leg length 5 cm Cy AA B C is also an isosceles right triangle 1339 What is the length of the hypotenuse What are the exact values of the sine cosine and tangent for AA What is the constant of proportionality from AABC to AA B C vnet 30 7 60 7 90 triangles 30 60 90 triangles Here are three of right triangles from the 30 7 60 7 90 set of triangles Calculate the missing angle measurement and the missing side length remember that Sketchpad is rounding the irrational numbers to two decimal places don t use rounded numbers like this if I ask for the exact value Are they similar to each other What s the scoop on the angle measurements if angles E A and P are 90 What are the constants of proportionality between the triangles Y E szYF3000O C P F YF 350 cm C mzCDP 30000 CD 600 cm PD 520 cm B A AC 100 cm BC 200 cm D mzABC 30000 Calculate the sin B cos B sin C cos C Will these be different in the other triangles What about the rounding off of the measurements sin Y sin D sin F sin C cos Y cos D cos F cos C Can you make a conjecture about 0 the relationship among all 30 7 60 7 90 triangles o the actual exact measures of the sides 30 60 90 triangle exercise What is the exact measure of side Con rm the exact values of the trigonometric functions of angles measuring 30 angle B and of angles measuring 60 angle A with the following triangle B A39 mzABA39 30000 AA39 200 cm AB 400 cm vnet chart summary with tangent Chart Summary Let s summarize what we know so far 7 and add a little bit more Here s a chart that summarizes the angles we ve worked on so far and adds in 0 and 90 D I we wig sin B 0 2 1 J3 5 1 cos B 1 Y 7 2 0 Here s an easy and automatic way to reproduce the chart Put the chart framework on the page Count off starting with zero left to right Count back starting with zero right to left Square root and divide by 2 B 0 30 45 60 90 sin B cos B Do it again B 0 30 45 60 90 sin B cos B Now we can discuss the tangents of all these angles Calculate the tangents of the famous angles B 0 30 45 60 90 i Q E s1n B 0 2 2 2 1 J3 5 1 cos B 1 Y 7 2 0 tan B You are expected to know these values by heart De nitions for all siX trigonometric functions If we have right triangle ABC and A is the right angle then 39d 39t B o the s1ne of angle B denoted s1nB or s1n B 1s the ratio w hypotenuse the cosecant of angle B denoted cscB or cscB is the reciprocal of sin B side adjacent B the cosine of angle B denoted cosB or cos B is the ratio hypotenuse the secant of angle B denoted secB or sec B is the reciprocal of cos B side opposite B the tangent of angle B denoted tanB or tan B is the ratio s1de adjacent B the cotangent of angle B denoted cotB or cotB is the reciprocal of tan B We will focus on sine cosine and tangent in the class It is however a good idea to have worked with the reciprocal functions a little bit Please fill in the following chart with the reciprocal function values vnet chart summary With reciprocal functions How about csc B sec B and cot B B 0 30 45 60 90 1 J5 5 sin B 0 2 7 7 1 Q Q 1 cos B 1 2 2 2 0 tan B csc B sec B cot B The rst 4 rows of information are to be memorized The next three are for general information and can be created at will vnet arbitrary right triangles Arbitrary right triangles Arbitrary right triangles don t have some special feature like two sides the same length or side lengths in proportion 3 7 4 7 6 they don t belong to nice tidy sets like the 30 7 60 7 90 right triangles They are scalene right triangles and they re not all similar Here s a right triangle and 4A is the right angle Note that it s a scalene right triangle that doesn t t any of the sets of right triangles we ve seen up to What is the exact measure of side CB What is the approximate measure of side CB sin B C cos C Why are these the same A B CA cos B 2 cm AB 6 cm sin C Why are these the same tanC tanB What is the relationship here vnet using the arckey Using the arcquot function key For this next part of the problem we ll need to undo the sine function going from a number that is the sin A to the measure for angle A We will use the function arcsin or sm 1 It is important that you know that the angles we are working with are acute angles happily this is always true with right triangles 7 why We ll get to how to handle obtuse angles in the section on the Law of Sines Let s practice this rst If A is an acute angle and sin A 82 what is the measure of A take arcsin 82 in Sketchpad or sin39182 in your calculator make sure your calculator is set on degrees You should get 5502 IfA is an acute angle and sin A 37 what is A A 2172 You can do this for the cosine function arccos cos39l and the tangent function arctan tan 1 too Now back to the problem we re working on from above CA20m AB60m nd the measure of A B nd the measure of A C 20 Arbitrarv right triangle exercise with arcsin work Given the following information nd the requested values Angle A is the right angle AB3cm AC7 cm What is the exact measure of the hypotenuse What is the approximate measure of the hyptenuse two signi cant digits Give exact measurements for these trig functions sinB sinC cosB cosC tanB tanC What is the measure of A B to two significant digits What is the measure of A C to two significant digits 21 An arcquot problem or two A mzA 90 What is the measure of A A in whole degrees What is the measure of A A A A What is the length of the third side of the triangle to two AHA 33946 Cm decimal places ie two signi cant digits A A 364 0quot What is tan A to 3 signi cant digits nd it using the de nition not the calculator tan key And another 7 the famous 3 7 4 7 5 triangle mzC 90 E C D EC 300 cm ED 500 cm What are the measures of the other two angles to two signi cant digits Use arccosine and arctangent functions just for the practice 22 Arcquot function exercise 1 All angles are acute angles Use the arcsin function arccosine function or arctan function SinA35 mzA CosB79 sz TanF25 sz SinC7l7 mzC CosD65 sz TanP258 mzP Arcquot function exercise 2 This is a right triangle hint tangent is a nice function F J3 What is the measure of angle C 23 Law of Sines and Law of Cosines Now what can we say about triangles that are not right triangles Well in some cases we can actually get some work done Let s talk about arbitrary triangles What is an arbitrary triangle It is a scalene triangle One that doesn t fit into a nice set like right triangles or isosceles triangles or equilateral triangles It has no nice features to make it like another triangle or triangles All three sides and all three angles have different measures One nice fact about an arbitrary triangle is that it can be decomposed into two right triangles easily Pick a vertex that is across from a side and run an altitude right to that side Recall an altitude is a line that connects a vertex to the side opposite or extension of the side opposite and is perpendicular to that side Be sure to pick a vertex and a side for which the altitude is interior to the triangle for decomposition illustration which altitude is the one we want 444444444 24 vnet Law of sines derived Decomposing an arbitrary triangle ABC Let s calculate some sines using hl and do a little algebra h l Now using h2 25 39 A 39 B 39 C Summarizing the algebra we have amp SH Sln a c This is called the Law of Sines for arbitrary triangles Given a triangle with vertices A B and C with sides opposite the respective vertices a b and c it is true that sinA sinB sinC a b c One caution at this time Sin C is not equal to sin A ACD sin A DCB You have to calculate it as the whole angle and not in pieces to be added back In other words sin C sin A ACD A DCB Add back the pieces rst and then calculate the sine not the other way around It is a pretty good law and has only one little problem There s an ambiguous case that you need to watch out for It turns out that the value sin A is the same as the value of another angle it is not an arbitrary relationship though and you can gure out which situation you have if you know about it We ll tackle this in our second Discovery Lesson ofthis section 26 Discovery Lesson Using the Law of Sines Open Sketchpad and sketch a triangle with angles that measure A 50 and B 35 Create a segment Use the Transformations menu and rotate successive by these measures Measure the angles using measure and have the measurements on the sketch Measure the side across from A BC 629 cm B A mzCBA 3500O mzCAB 50000 Using Calculate and selecting the sine function from the drop down menu calculate the Sines of the angles by rst selecting sin and then clicking on the angle measure then click ok 0 BC 629 cm B A mchA 35000 mchB 50000 sinmCBA 057 simmZCAB 077 27 39 A 39 B Now us1ng the formula Sln solve the formula for side b Again using the Calculate function calculate the length of side b Use the distance function to check your calculation What did you nd Construction notes and tips 28 Law of Sines continued Exercise using the Law of Sines Use your calculator or Sketchpad calculator to solve the next problem The measure of angle A is 49 degrees The measure of angle Bis 37 degress The length of side CA is 25 inches What is the length of side BA Also what is the measure of angle B 29 Discovery Lesson The Ambiguity 0f the Arcsine Function with Arbitrary Triangles Use your calculator or the calculator in Sketchpad for this lesson Here are two angles Calculate the sine of each D mLBAD 145000 szAC 35000 B A C What do you notice What is the relationship of the angles Now take arcsin of the sine value Do you get two answers Here are two triangles C 4 C39 D I g 1 A B A39 B39 mCBA14000 miA39B39D 4000quot Calculate the sine of 4B and AB What do you notice What is the relationship of the angles Now take arcsin of the sine value Do you get two answers Which angle do you see on the calculator face 30 Here is a sine value sin A 79 Take arcsin 79 or sin 391 79 Is this the only value for A nope What s the other angle that has sine of it is 79 How is it related to the rst angle Here s a problem BD1121cm DA 371 cm szBA1612 Use the Law of Sines to solve for sin DAB and the arcsin function to get m4 DAB What happens How do you x it 31 More Law of Sines problems use two or three decimal places AC 1088 cm BA 550 cm szCA 2139O What is the measure ofangle ABC another one BA519cm A026 Cm szCA 4676O B C What is the measure ofangle ABC 32 vnet Law of Cosines Law of Cosines The Law of Cosines is very handy too and it is for arbitrary triangles In words The measure of a side of a triangle is the square root of the quantity the sum of the squares of the other two sides less twice the product of those sides and the cosine of the enclosed angle There is no ambiguous case for the Law of Cosines If you happen to use it to solve for an angle measure and we will you may take the answer that comes out of the calculator at face value Let side AB be the side we care about AB AC2 CB2 2ACCBcos C Three illustrations of the Law of Cosines What is the length of side AB rounded to two decimal places C AC 537 cm CB 353 cm mzACB 72900 33 Another problem What is the measure of A B to two decimal places gt 0 AB 552 cm BC 421 cm AC 691 cm And another one give your answer rounded to two decimal places AC 647 cm B BC 6 32 C 39 39 cm mBCA14130 What is the measure of side AB 34 3 4 5 Law of Cosines exercise Find the measure of the angle across from the 3 cm side 4B in the standard 3 7 4 7 5 triangle using the Law of Cosines 35 Area of a triangle vnet area ofa triangle derived What is the formula for the area of a triangle We can change this to something equally true and perhaps just as useful using decomposition and trigonometry Here s how Here s an arbitrary triangle decomposed into two right triangles by the altitude CD The usual area formula is base height where height is h the length of CD and the base is the length of side BC C We will say AACB A 9 for simplicity 36 Now we know from the Law of Sines that a This means that sin A asm 9 AB We also can see from the de nition of sin A that sin A This means that h b sin A Now let s take our original area formula 1 Area ABh EABbsin A me ABb 7 absinG AB 1 l 2 2 So the area of a triangle is the one half the product of two sides and the enclosed angle Let s look at some applications of this new way of calculating area vnet area of an equilateral triangle Area of a re ular hexa 0n Suppose we have a regular hexagon with the distance 1 cm from the center to each vertex What is the area of the hexagon and multiply by 6 we can get the area of the whole B If we calculate the area of one subunit 7 the triangle A hexagon PB100cm What is the measure of A A What is the exact area of the triangle What is the exact area of the hexagon 37 Area of arbitrarv triangles 1 and 2 Find the areas to two decimal places A AC 647 cm B C BC 632 cm szCA 141 300 What is the area oftriangle ABC A Problem AB 733 cm AC 604 cm BC 632 cm What is the area of triangle ABC 38 Wrapping it up exercise Come up with as many different ways as you can to nd all the side lengths and all the angle measures for this triangle What is the area of the triangle Does the old formula answer match the new formula answer What accounts for this 39 Unit Circle Trigonometry Since the advent of the function as a mathematical concept the study of trigonometry has moved away from triangle applications to a more modern approach The ratios we studied earlier are actually functions 7 they pass the vertical line test when graphed We will study a method of ascertaining the Cartesian coordinates for the 3 trigonometric functions that we have studied so far Pythagorean Identity First we will look at the famous Pythagorean Identity vnet Pythagorean Identity Take a right triangle and pick one of the acute angles say B Now assign the side names relative to B and apply the Pythagorean Theorem hypotenuse adj opp2 adj2 hyp2 Now suppose we use algebra opp2 adj2 hyp2 hyp2 2 2 Ld 1 sin2 Bcos2 B 1 hyp hyp 1 40 The Coordinate Connection vnet connection to Unit Circle We can also 7 being clever with algebra again 7 turn the identity into a geometric shape Let X cos B and y sin B mnemonic both are in alphabetical order X y and cosB sinB This gives us the formula for the unit circle 7 a circle of radius 1 centered at the origin Now suppose we set up our coordinate system inside a unit circle and look at what we can build We ll have a device for cranking out lots of values for sine and cosine all in one little piece Take a unit circle Using the X aXis as the initial ray and a radius of the circle as a terminal ray rotate in either direction as many times as you wish Each time you stop you ve got a value for 9 the rotation Counterclockwise rotation is a positive rotation and clockwise is a negative rotation Note that the tip of the radius is on the circle The X coordinate is associated with cos 9 and the y coordinate is associated with sin 9 A counterclockwise rotation is positive w ms s sin e 41 vnet angle rotations Where is 0 45 720 120 312 180 Let s look at this new idea that X cos 0 and y sin 0 Let s go around the circle marking in signs for sine and cosine This is a new idea with triangles all the trig functions are positive but that is not true in our new expanded View 42 Quadrantal Angles vnet quadrantal angles Now let s look at Quadrantal Angles 0 90 180 and 270 What are the point values for 0 Convert these into trig functions sin 0 cos 0 tan 0 Now 90 point coordinates sin 90 cos 90 tan 90 Now do 180 and 270 on your own in the homework Hint look closely at 90 for relationship with 270 and look at 0 for relationship to 180 43 Reference Angles There are quite a few more angles we will cover All of these will have to be learned by heart and the graphs should be by heart too To start off let s rewrite our chart of reference angle values angle 30 45 60 90 sine cosine tangent In rotating around the unit circle the pointer ray touches points that have the same trigonometric function values as the above well know angles in the first quadrant Note for example how 30 and 150 are symmetric about the yaXis If you fold the Unit Circle the two rays will be right on top of one another In absolute value then the coordinates of the points are the same They differ with respect to the sign plus or minus 44 vnet symmetry and reference angles Let s look at the theta for the coordinates 6 8 Check to make sure these actually ARE coordinates on the Unit Circle Use arcsin8 to discover the measure of the rotation What triangle does this come from w an s sin e Look in Quadrant two which angle has similar coordinates the same absolute value but with differing signs Quadrant 3 Quadrant 4 What conclusion can we draw 45 vnet angles referenced by 30 Let s start with 30 It is a reference angle for 3 other angles when considering one rotation of the terminal angle Why can it serve as a reference Let s look at these What are the trig function values for 30 Now let s talk values for the trig functions 150 210 330 46 vnet angles referenced by 45 Now let s look at 45 It s the reference angle for the following 3 angles 135 225 and 315 why What are the trig values for 45 Now let s talk about 135 225 315 aka 45 why 47 vnet angles referenced by 60 And nally for 60 The related angles are What are the trig function values for 60 now for 120 240 aka 120 why 300 48 If you allow more than one rotation of the terminal leg of the angle then you can start talking about angles that measure more than 360 or have negative measure It will be important to identify the reference angle and the Quadrant then the trig function values are easy vnet reference angle exercises What are the exact values for the following Reference angle exercise give the reference angle and the exact value sin 60 tan 225 cos 150 sin 210 sin 405 49 tan 120 cos 270 sin 90 sin 90 cos 225 tan 90 sin 300 50 vnet periodicity Periodicity 7 once you ve gone from 0 to 360 you begin repeating the trig function values with ever higher numbers for the rotations And if you rotate in a negative fashion you cycle through the same values as well sin A sin A i 360 cosA cos A i 360 however tan A repeats with a period of 180 Periodicity exercise nd the values remember to use references when necessary sin 585 cos 540 tan 750 cos 420 sin 570 51 A review of t 39 ref 4 and ic function values exercise sin 9 cos 9 52 vnet wrapping function Now we want to move completely into algebra and graph the trigonometric functions We ll do the sine function and the tangent function in class and leave the cosine function for homework The rst matter to deal with is the fact that the unit circle visualization of trigonometry is not a function at all it doesn t pass the Vertical Line Test It is a convenient device that gives us all the information that we need to make the 3 graphs that we want though We want the XaXis to be a linear representation of the rotation 9 In order to achieve this we have to look hard at the relationship between circles and number lines If you snap a number line onto a circle with the number zero right at a rotation of zero you can begin looking at this relationship Choose a positive rotation 9 and imagine rolling the sphere on the number line until the circle point that de nes the point of intersection of theta s ray is touching the number line We ll call the distance from 0 to that point a linear representation of the angle theta Now do it with a negative rotation Notice that using the actual degree measurement for the number line could work if we set up a correspondence between 1 and a unit of linear measure most students want this to be true too There is however a more conventional unit measure that ties in other facts about rotations better than this Our unit of measure will be 1 radius length 1 linear unit 53 Extend the number line part of the following sketch til it lls the whole paper Worksheet One unit length a radius Use your pipe cleaner to measure off the radians on both the circle and the axis 54 Do the integers rst Then we ll discuss where 180 and 360 et al Let s spend some time on the idea of using a radius as a unit of measure Ok so what s the value of 180 in linear units Why is this true What s the circumference of a circle Isn t this what we just did on the preceding page What s the numerical value of pi rounded to two digits So there would be some dissonence with haVing 314 coming smack in between 179 and 180 on a number line which rules out a nice 11 correspondence between degrees and numbers on the numberline Using a radius is the only way to get pi placed correctly on the number line 55 This number line we ve made will be the XaXis that we ll use when we graph the trigonometric functions There is a handy way to get the XaXis values using a conversion factor 11 180 Rewrite the aXis values from 211 to 211do the quadrantal angles first and then the 4 angles below Number line exercise Convert each measurement from degrees to radians or vice versa Fill in the following rotations on the number line above 51 12 50 80 711 12 511 75 315 31 4 210 39 l 311 20 120 66 56 Now let s get to graphing First we ll update our chart to include more quadrantal angles and radian measure for the rotations B 0 30 45 60 90 180 270 radian measure sin B cos B Q2 related rads Q3 related rads Q4 related rads tan B later What s the largest value in the table What s the smallest value in the table The sine function 57 vnet graphng the sine function Worksheet working with the paper turned sideway ll in the points for the sine function From 211 to 211 58 Let s go back on our graph and mark off the quadrants for positive rotations for one period one rotation 59 The sme funcuon m n39s mostbasxc single penod Now let s graph the tangent function vnet graphng the tangent function for two pen39ods go back and ll in the cha1t values 61 The tangent funcuon Answers to Exercises Similar triangles Isosceles right triangle 30 7 60 7 90 triangle Arbitrary right triangle with arcsin work An arc problem or two Arc function 1 Arc function 2 Exercise using the Law of Sines Corresponding angles are congruent and corresponding sides are proportional Ato B 43 B to A 3A 1 5J2 same same same 53 5 a 3377 5 2J3 sin 30 cos 60 05 cos 30 sin 60 1 tan30 3 2 tan60 3 sinBcosCL J58 58 cosBsinC inanB 1 J58 3 tan C is reciprocal tan B B E 6680 arcsin9l9l4503 C 90 6680 232 mzA arcsin346364 5 72 mzA 180 790 772 18 113 tanA m 327 3687 and 5313 2949 3781 1404 560122 mzC 180 7 49 7 37 94 sin 37 sin 94 25 AB AB 414 63 More Law of Sines Problems Three illustrations of the Law of Cosines s1n 2139 s1nB Sian 721 55 1088 B arcsin 721 m 4614NOTit s obtuse more than 90 B 180 arcsin721 13386 second one is acute can use the calculator answer directly sin B sin 4676 672 519 sin B m 943 B is arcsin 943 aka sin391943m 7056 rst AB 5372 3532 2537353cos 7290 AB m 549 cm second note that we are solving for the cosine of B and then using the cos391 or arccos key in Sketchpad or the calculator 691 5522 4212 2552421 cos B square both sides subtract and diVide down to cos B m 00960446 take arccos both sides B m 8943 third AB 6322 6472 2632647cos14130 AB m 1207 cm 64 3 7 4 7 5 Law of Cosines Area of a regular hexagon Area of arbitrary triangles l and 2 Wrapping it up exercise 1 Reference angle exercise 3 45 42 254cosB 32 52 42 cos B 8 25X4 arccos 8 cos3918 m 3687 E B 36 60 1l lsin6 7 67 2 4 4 2 Triangle l 12 632647sinl4l30 m 1278 sqcm Triangle 2 need to nd an angle measure rst Use Law of Cosines to nd that mzA m 554l Area 12 733604sin5541 m 1822 sqcm 53 x 3795 cm y 2155 cm 73 sin 60 60 QIV 7 tan 225 45 QIII 1 cos 150 30 QII sin 210 30 QIII 12 f sin 405 45 Q1 72 tan 120 60 5 cos 270 quadrandal 0 sin 90 quadrantal 1 sin 90 quadrantal l 65 Periodicity exercise 5 cos 225 45 QIH 7 tan 90 quadrantal unde ned 5 sin 300 60 QIV 7 J5 sin 585 sin 225 360 45 amp QIII 7 cos 540 cos 180 360 quadrantal 1 tan 750 tan30 2360 g cos 420 cos 60 360 cos 60 12 sin 570 sin210 360 30 amp QIII 12 66 A review of trigonometric function values exercise 0 ref and sin 0 cos 0 tan 0 Quad rant 0 quadrantal 0 1 0 30 Q1 12 Z 3 45 QI z Z 1 60 Q1 432 12 5 90 quadrantal 1 0 unde ned 120 60 QII z 12 5 135 45 QII z 2 1 150 30 Q11 12 432 3 180 Quadrantal 0 1 0 210 30 QIII 12 z 3 225 45 QIII E2 JE2 1 240 60 QIII z 12 270 quadrantal 1 0 unde ned 300 60 QIV z 12 5 315 45 QIV 2 z 1 330 30 QIV 12 52 53 360 quadrantal 0 1 0 30 30 QIV 12 z 3 390 30 Q1 12 52 3 45 45 QIV z z 1 405 45 Q1 z Z 1 67 60 60 QIV 2 12 420 60 QI z 12 Number line exercise Do the conversions Now put them in ascending order smallest to largest then put them left to right on a horizontal number line Each of these has a sine value a cosine value and a tangent value which can be graphed in point pairs 210 7l 120 31 fl 100 9 80 7 9 1 7 31 675 24 8 39o J37 60 1 210 18 24 7 E 30 31 27 20 500 511 18 66 m 30 31 675 8 51 75 12 71 105 12 68 311 7 270 4 315 71 4 1L 3400 9 3960 5 69
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