SL1.Trig.1718.notebook. April 15, /26 End Q3 Pep talk. Fractal Friday: April 6. I want to I will I can I do

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1 Coming up Explorations! A few ideas that I particularly like: Complex quadratics Fractals (complex numbers) Graphing complex roots of quadratics Geometric interpretation of variance etc. Understanding covariance Multiple regression, Galton, Pearson Dot product as correlation! A two variable box & whisker? Test Newton's law of cooling Matrix transformations in animation...many others on website Topic due here Draft due here Final due here Fractal Friday: April 6 I want to I will I can I do Discuss Probability Exams Quick Quiz 2/26 End Q3 Pep talk 1

2 Spring Break Overview 2

3 Right Angle Trig Review Geogebra Sketch to Explore Lots of ways to use these ideas! 2/26Rt Angle Trig Review 3

4 1. Understand and use radian measure 8A: #1ejo,2de,3hij,4cde,5 (Radians) Handout: Right Angle Trig Review Lots of theories about where 360 degrees came from. The Babylonians has a base 60 number system that may have contributed. One version of a Mayan calendar used 20 cycles of 18 days (360) plus 5 unlucky days! The Persian calendar used 360 days for a year. Note the connection to hours, minutes and seconds time was initially measured based on astronomical cycles that were assumed to be circular. Roger Cotes is generally credited with defining a new unit of measure. The radian. It was natural to measure an angle by the length of the arc that it subtends. But that length would be different for different size circles. Unless, of course, you measure the length in "numbers of radii of the given circle". Thus the name "radians". α = 1 radian So how do we convert between the systems? How many radians (radii) are there in a complete circle? Distance around a circle is 2πr so: 360 degrees = 2π radians or 180 degrees = π radians Try a couple: Pay close attention to #5. Look for a recognizable pattern. 8A: #1ejo,2de,3hij,4cde,5 (Radians) Handout: Right Angle Trig Review RightAngleTrigReview.PDF 2/26 8A Radian Measure 4

5 8A: #1ejo,2de,3hij,4cde,5 (Radians) Present as needed: #5 by memory! Handout: Right Angle Trig Review 1. Understand and find arc length & sector area. 8B: #2 12 even (Arcs & Sectors) QB: #3,10,13,20 (Arcs & Sectors) First, some vocabulary Derive a formula from your understanding of circles. Derive a formula from your understanding of circles. 8B: #2 12 even (Arcs & Sectors) QB: #3,10,13,20 (Arcs & Sectors) 2/27 8B Arc Length and Sector Area 5

6 8B: #2 12 even (Arcs & Sectors) Present 6,8,10,12 QB: #3,10,13,20 (Arcs & Sectors) Present all 1. Understand trig functions as locations on the unit circle. 8C: #1b,2 9 (Trig on the unit circle) QB: #27,33,37,77 (Arcs & Sectors) A circle of radius one is one of the most fundamental geometric shapes. It is a very effective visual reference for many ideas involving trigonometry. What is the equation of a unit circle? (Hint: What is a circle?) The set of points equidistant from a center x 2 + y 2 = 1 for a radius of 1 Let's look at a point on the unit circle, located at an angle θ measured from the +x axis. Unit Circle Trig Tracers From this we see some important principles: For a point on the unit circle: These are more general definitions than SOHCAHTOA since they include negative values and angles greater than 90 deg. This is a more general definition of tangent. Notice that if we add 2π radians to any angle, θ, we have taken one full revolution around the circle and end up at the same position. Thus the trig functions are periodic. This is true for adding any integer multiple of 2π radians to θ we are simply revolving around the circle an integer number of times (in either direction!) Mathematically speaking, For θ in radians and k Z, sin(θ + 2πk) = sin(θ) and cos(θ + 2πk) = cos(θ) Tangent is a little different. Because it is a ratio of sin to cosine, it repeats every half revolution: For θ in radians and k Z, tan(θ + πk) = tan(θ) Pay close attention to the results of #2. You will want to use this opportunity to memorize (re memorize?) these results. 8C: #1b,2 9 (Trig on the unit circle) QB: #27,33,37,77 (Arcs & Sectors) 3/1 8C Unit Circle 6

7 Triangle Trig Quiz Thursday A big week Know your angle identities, inverse trig functions, unit circle and Pythagorean Triples 8C: #1b,2 9 (Trig on the unit circle) Present 3,4,5,8,9 QB: #27,33,37,77 (Arcs & Sectors) Questions as needed 1. Work with trig functions on the unit circle. 8D.1: #1 5cd,6 (More Trig on the Unit Circle) 8D.2: #1adg,2adg (Inverse Trig) Summary of trig relationships so far: The Big Kahuna: sin 2 α + cos 2 α = 1 Supplementary Angles: sin(π α) = sinα cos(π α) = cosα tan(π α) = tanα Complementary Angles: sin(π/2 α) = cosα cos(π/2 α) = sinα tan(π/2 α) = cotα Negative Angles: sin( α) = sinα cos( α) = cosα tan( α) = tanα π plus an Angle: sin(π + α) = sinα cos(π + α) = cosα tan(π + α) = tanα π/2 plus an Angle: sin(π/2 + α) = cosα cos(π/2 + α) = sinα tan(π/2 + α) = cotα 2πk plus an Angle (k Z): sin(2πk + α) = sinα cos(2πk + α) = cosα tan(πk + α) = tanα Do not try to remember these. Just know that they exist. When you see these related angles, construct the relationship in your head by visualizing the unit circle! Try some applications: We can use algebra and the unit circle or "directed triangles" 3 θ 2 1. Draw a triangle with the known ratio 2. Use Pythagorus to quickly find the third side 3. Identify the other ratios from the triangle 4. Remember that there are two answers: ± 2 θ 3 Notice that trig functions are not one to one! For example, the sin or cosine of an angle is a single value, but there are two angles between 0 and 2π that have a particular sin or cosine! Pay attention to domain constraints The domain constraint may lead to a single value for your answer. Consider the previous examples What were we given? (the value of a trig function for some angle a ratio) What were we trying to find? (the value(s) of another trig function for the same angle) We made use of the Pythagorean Identity: sin 2 θ + cos 2 θ = 1 It's a huge idea in trig always look for it! Try another one: Pay attention to domain constraints In this example What are we given? (the value of a trig function for some angle) What are we trying to find? (the angle(s) that have the given trig function value) To do this we use the inverse trig function generally on your calculator (ex: sin 1 ) But remember that trig functions are not one to one. For example there are lots of angles ( actually) with a sin of 0.5. Name some: π/6, 13π/6, etc. 5π/6, 17π/6, etc. The full set is: π/6 + 2πk and 5π/6 + 2πk So why do I call it an inverse trig function? Inverse Trig Functions Inverse trig functions are defined only for a limited domain: To find all the angles that satisfy the conditions of the problem, we must do two things: 1. Identify the domain constraints 2. Use the following identities to find all the angles that meet the constraints For θ in radians sin(π θ) = sin(θ) cos(π θ) = cos(θ) cos(2π θ) = cos(θ) You discovered this in your HW from 8C There are lots of these relationships. Understand where they come from so you can reconstruct them as needed. Do not try to memorize these formulas! Let's try one more: 8D.1: #1 5cd,6 (More Trig on the Unit Circle) 8D.2: #1adg,2adg (Inverse Trig) Triangle Trig Quiz Thursday 3/5 8D Applications of Unit Circle 7

8 8D.1: #1 5cd,6 (More Trig on the Unit Circle) Present #1 5 c or d, 6 8D.2: #1adg,2adg (Inverse Trig) Present #1g,2g Mini Quiz: [2] each Triangle Trig Quiz next Time Know your unit circle and Pythagorean Triples 1. Understand the origins of and memorize exact trig function values for common angles. 8E: #1 4,5cfil,6 10 (Common angles) Unit Circle Handout 8F: #1 2 (Equations of lines) Review Set 8C: #1 12 as needed (Review) Find, showing all your work, the exact values for 1. Sin, cos, and tan of 30 and 60 (Hint: Start with an equilateral triangle) 2. Sin, cos, and tan of 45 (Hint: Start with a square) Use the values you found to fill in the unit circle on the handout with the requested information. Keep it for your records. Memorize it. We will have a quick quiz on this next time and a larger quiz on all trig ideas so far on Tuesday. What about triangles that are Pythagorean Triples? They have rational values for all the trig ratios. The quiz on Thursday will require that you memorize the first four prime Pythagorean Triples. They are: Knowing these four triples will save you a huge amount of time as they are so often used. Do any of them have nice (ie. integer) angles? Check on your calculator. Challenge: Find the smallest Pythagorean Triple that has integer values for the angles in a right triangle...or prove that such a triple does not exist. 3/6 8E Common Angles 8

9 1. Recognize and use the tangent as the slope of a line Find the slope of the line shown. Hence, find its equation. b θ Triangle Trig Next Time Know the unit circle and Pythagorean Triples! 8E: #1 4,5cfil,6 10 (Common angles) Unit Circle Handout 8F: #1 2 (Equations of lines) Review Set 8C: #1 12 as needed (Review) 3/6 8F Straight lines 9

10 Return and discuss Quiz on Ch 8 Arc Length, Sector area, unit circle. 1. Understand how use trig to find the area of a triangle from two sides and the included angle. 9A: #1 11 odd (Areas of triangles) We can use trig to calculate the area of a triangle. First, consider the acute triangle below: We know area to be A = ½ha But notice that sinc = h/b so h = b sinc Thus, A = ½ a b sinc We call C the included angle between a and b. What about obtuse triangles? Again A = ½ha But now C is obtuse and our right triangle has an angle of 180 C. We've already seen that sin(180 C) = sin C = h/b so it's still true that h = b sinc And, again, The idea can also be used in reverse to find an angle, given an area and two sides: Notice two possibilities! 9A: #1 11 odd (Areas of triangles) 4/9 9A Area of a Triangle 10

11 9A: #1 11 odd (Areas of triangles) Present 3 11 Mini Quiz: Find the exact shaded area Find area of sector = ½(64)3π/4 = 24π Find area of triangle = ½(64)sin(3π/4) = 32* 2/2 = 16 2 Subtract triangle from sector = 24π 16 2 M1A1 M1A1 M1A1 8 mm 1. Use the cosine rule to find angles and sides of non-right triangles. 9B: #1b,3,5,6c,8a,9 (Cosine Rule) QB: #1 & pick 2 from 4,5,14,17,19 (Cosine Rule) Consider the triangle at right: Note that h can be computed from ΔADC or from ΔBDC Thus h 2 = b 2 x 2 = a 2 (c x) 2 Let's simplify: b 2 x 2 = a 2 (c 2 2cx + x 2 ) b 2 x 2 + (c 2 2cx + x 2 ) = a 2 b 2 x 2 + c 2 2cx + x 2 = a 2 b 2 + c 2 2cx = a 2 But trig tells us that x = bcosa Substitute to get b 2 + c 2 2bccosA = a 2 (Pythagorus) (Definition of cos) The cosine rule The Law of Cosines (Cosine Rule) For a triangle with sides of length a, b, & c whose opposite angles are given by A, B, & C respectively, the following relationships hold: Notice that the angle is the included angle between the two known sides. If the non included angle is given, the situation is ambiguous as the resulting equation will be quadratic, with two potential solutions. Consider: 6 2 = c 2 2(10)c cos30 36 = c 2 20c( ) 0 = c c + 64 So c = or 5.34 The good news is that using the cosine rule will make it clear that there are multiple solutions because the equation will tell you that! Some rearranging gives some other useful forms: The book work is a warm up for the QB problems. Do 6,8 & 9 minimum. Focus on the QB. 9B: #1b,3,5,6c,8a,9 (Cosine Rule) QB: #1 & pick 2 from 4,5,14,17,19 (Cosine Rule) 4/10 9B Cosine Rule 11

12 Folk Art Market July Quiz on Ch 9 Next Thur 4/19 9B: #1b,3,5,6c,8a,9 (Cosine Rule) Present #3,5,9 QB: #1 & pick 2 from 4,5,14,17,19 (Cosine Rule) Present #4,5,14,17,19 1. Use the sine rule to find angles and sides of non-right triangles. 9C.1: #1b,2b (Sine Rule) 9C.2: #1,2b,3,5,8 (Ambiguous Case) QB: #22,39,49,68(3D) (Sine Rule) We have shown that the area of a triangle can be given as: ½ab sinc = ½ac sinb = ½bc sina By multiplying all three expressions by 2 and dividing them all by the product abc we get the sine rule. The Law of Sines (Sine Rule) For a triangle with sides of length a, b, & c whose opposite angles are given by A, B, & C respectively, the following relationships hold: You can use the sine rule to find sides or angles for non right triangles. Some cases are straightforward. 4/12 9C Sine Rule (sides) 12

13 Now draw a triangle with a = 3, b = 5 and A = 30 and solve the triangle using the sine rule. Did you draw this? 3 Did anyone draw this? The Law of Sines gives you exactly the same result! How can this be? The answer lies in the fact that there are two angles < 180 that have a sin of 5/6. One is which your calculator provides. But the other is = So angle B might also be ! Now notice the given angle, 30. If we add it to we end up less than 180 so there is still 180 ( ) = left for angle C. Thus we have a second solution! The Ambiguous Case of the Law of Sines When given two sides and a non included angle, it is possible two have one, two or no solutions to the triangle! You must explore both solutions to sin 1. Using the cosine rule will help! Do more of 1&2 if you need practice. Quiz on Ch 9 next Thur, 4/19 9C.1: #1b,2b (Sine Rule) 9C.2: #1,2b,3,5,8 (Ambiguous Case) QB: #22,39,49,68(3D) (Sine Rule) 4/12 9C Sine Rule Ambiguous Case 13

14 9C.1: #1b,2b (Sine Rule) 9C.2: #1,2b,3,5,8 (Ambiguous Case) QB: #22,39,49,68(3D) (Sine Rule) Present all Present all Quiz on Ch 9 This Thur, Apr Use the sine and cosine rules in context. 2. Review use of bearings and angles of elevation and depression. 9D: #2 16 even (Using Sin & CosRules) or QB: #43,46,53,60,73 (Cosine Rule) QB: #32,50,57 (Sine Rule) In problems involving solving triangles, you have three options: > Try to find right triangles and use definitions > Use the cosine rule > Use the sine rule Often more than one approach can work. Choose the one that is easiest. Bearings: The clockwise angle measured from North. Use other problem solving skills sketches, naming variables, writing down "raw" equations, break the problem into simpler steps, etc. Quiz on Ch 9 This Thur, Apr 19 9D: #2 16 even (Using Sin & CosRules) or QB: #43,46,53,60,73 (Cosine Rule) QB: #32,50,57 (Sine Rule) 4/16 9D: Applications 14

15 Quick Quiz: Given the proportion find the exact value of x, giving your answer in the form x = a + b c where a & b and c 9D: #2 16 even (Using sin & cos rules) Present all Review Set 9C: #1 6 (Review) QB: #43,46,53,60,73 (Cosine Rule) Take Home Quiz Ch 9 Today we will 1. Review and begin our exam. Review Set 9C: #1 6 (Review) QB: #43,46,53,60,73 (Cosine Rule) Take Home Quiz Ch 9 4/9 9D: Applications 15