Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 1 Chapter 8A Angles and Circles From now on angles will be drawn with their vertex at the The angle s initial ray will be along the positive. Think of the angle s terminal ray as starting along the positive x-axis, and then swinging into its position. If the terminal ray swung away from the x-axis in a counterclockwise direction, then the angle has measure. If the terminal ray swung away from the x-axis in a clockwise direction, then the angle has measure. The circle below has a radius of 1 unit. It is called the. The circumference of a unit circle is If a terminal ray swings through an entire rotation, you would say it has a measure of You could also say that it has a measure of.
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 2 1. Sketch the following angles on the unit circle below a) π 4!! b) 2π 3!! c) 7π 6! d) 3π 2! e) 11π 12 2. Sketch the following angles on the unit circle above a) π 4!! b) 2π 3!! c) 13π 6! d) 3π! e) π 3π and π are called because they share the same To find an angle that is coterminal to θ, just add or subtract Another way to say this: To find an angle that is coterminal to θ, just add or subtract 3. List 2 other angles that are coterminal angles with π 2 4. List 2 other angles that are coterminal angles with 2π 3 If θ 1 and θ 2 are coterminal angles, then θ 1 -θ 2 =
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 3 An angle is called acute if its measure is between An angle is called obtuse if its measure is between Two angles are called complementary if the sum of their measures is An example of complementary angles: θ 1 = and θ 2 = Two angles are called supplementary if the sum of their measures is An example of supplementary angles: θ 1 = and θ 2 = A line which intersects the circle twice is called a A line which intersects the circle at exactly one point is called a The region inside of a circle is called a Any piece of the circle between two points on the circle is called an Any line segment between 2 points on the circle is called a Any piece of the disk between 2 radial lines is called a An angle whose vertex is at the center of a circle is called
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 4 Three ways to measure angles: Revolutions -- Degrees -- Radians Revolutions Degrees Radians 3 2 3 1 3 75 480 5π 5π 12 Arc Length:
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 5 Area of a sector: When using degrees to measure the central angle! Area of a sector =!!!!!! Length of arc =! When using radians to measure the central angle! Area of a sector =!!!!!! Length of arc =!
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 6 Suggested Problems: Text: 1-12 My Previous Exams:!! S14 1A: 5,! S13 3A: 11!! F12 3A: 9, 11 Dr. Scarborough s Previous Exams: F13 III: p2:4 F12 III: p4:9 Dr. Scarborough s Fall 2013! WIR 9: 2, 3, 8-10, 13-15, 43 Dr. Kim s Fall 2014 WIR:
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 7 Chapter 8B - Trigonometric Functions Recall from geometry that if 2 corresponding triangles have 2 angles of equal measure, then they are. E C A D B Since ABC and ADE are both right triangles sharing the common angle A, they are similar triangles. When 2 triangles are similar, it means the lengths of their corresponding sides are proportional. So!! AD AB = These ratios can be rewritten as BC AC =!!!! and as! BC AB =
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 8 This means that given a right triangle with an angle measured θ, the ratios are constant. There are six functions that can be defined with θ as their argument. sinθ cosθ tanθ!!!!!!!! cotθ cscθ!!!!!!!! secθ Tom s Old Aunt! Sat On Her!!! Coffin And Howled Notice that sinθ cosθ!!!!!!!!! cosθ sinθ 1 cosθ!!!!!!!!!!! 1 sinθ
Consider the equilateral triangle sin π 3 cos π 3 tan π 3 csc π 3 sec π 3 cot π 3 sin π 6 cos π 6 tan π 6 csc π 6 sec π 6 cot π 6 Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 9
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 10 Consider the isosceles right triangle sin π 4 cos π 4 tan π 4 csc π 4 sec π 4 π cot 4 The trouble with these definitions of our trigonometric functions is that they are defined only for 0 < θ < π 2.
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 11 Another way to define the trigonometric functions: Given a circle of radius one, (often called the unit circle) with a radial line drawn at an angle θ, measured counterclockwise from the positive x- axis, the radial line intersects the circle at a point (x, y). The trigonometric functions can then be defined as sinθ cosθ tanθ cscθ secθ cotθ Find cos 3π 4
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 12 Let θ be an angle in standard position. The reference angle θ is the acute angle formed by the terminal side of θ and the x -axis. cos 5π 6 sin 4π 3 sin π 6 tan π 2
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 13 Reference Angle Theorem: Let trig( θ ) be any one of the six trigonometic functions defined above (on page 26).!!!! Then trig( θ ) = ±trig( θ )!!!! The correct sign is determined by the quadrant of θ. tan 7π 6 csc 5π 3 1. Given that sinθ = 3 5 and θ is acute, determine the values of a) cosθ!!!! b) tanθ!
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 14 2. Given that sinθ = 3 5 and θ is NOT acute, determine the values of a) cosθ!!!! b) tanθ! 3. Given that tanθ = 2 and cosθ < 0 determine the value of sinθ.
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 15 4. Given that secθ = 7 5 and tanθ < 0 determine the value of tanθ. Suggested Problems: Text: 3-11 My Previous Exams:!! F13 3A: 6,!! S13 3A: 12,!! F12 3A: 13 Dr. Scarborough s Previous Exams:! F13 III p3:5, p8:8!!!!!!!!! F12 III p3:4 Dr. Scarborough s Fall 2013 WIR 9: 5, 17, 19, 23, 25, 35, 41, 44, 51 Dr. Kim s Fall 2014 WIR:
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 16 Chapter 8C Graphs of Trigonometric Functions sinθ = 0 when θ = In other words sinθ = 0 when θ = sinθ = 1 when θ = In other words sinθ = 1 when θ = sinθ = 1 when θ = In other words sinθ = 1 when θ = sinθ = 1 when θ = 2 In other words sinθ = 1 2 when θ = Use this information to carefully plot y = sinθ for 2π θ 4π What is the domain of y = sinθ! What is the range of y = sinθ?
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 17 Amplitude!! Plot y = 3sinθ, comparing it to y = sinθ What is the range of y = 3sinθ? Consider the graph of y = 2sinθ Notice that when θ = π 2, y = 2. y = 2sinθ is a reflection of through the The amplitude of y = 2sinθ is. The range of y = 2sinθ is The function y = asinθ has an amplitude of. What is range of y = asinθ?
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 18 Vertical Shift! Plot y = (sinθ) 2 comparing it to y = sinθ Here the graph is oscillating about What is the amplitude of y = (sinθ) 2? What is the range of y = (sinθ) 2? Period The sine function is considered periodic, because sinπ = sin 3π ( ) and sin π 4 = sin π 4 + 2π. In fact, sinθ = sin ( θ + 2π) for all θ. The period of y = sinθ is
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 19 Complete the table and plot y = sin2θ and y = sin θ comparing it to y = sinθ 4 θ 2θ y = sin2θ θ θ 4 y = sin θ 4 0 0 0 0 π 4 π 2 2π π 2 π 2 π 4π π 3π 4 3π 2 6π 3π 2 π 2π 8π 2π 2π π 2 The period of y = sin2θ is The period of y = sin θ is 4 The period of y = sin kθ is If y = sin2θ, then k =, so the period of y = sin2θ is If y = sin θ 4, then k =, so the period of y = sin θ 4 is If y = sin 3θ, then k =, so the period of y = sin 3θ is
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 20 Phase Shift Plot y = sin(θ π ) and y = sin(θ + π ) comparing it to y = sinθ 4 2 θ θ π 4 π 4 3π 4 5π 4 7π 4 9π 4 y = sin(θ π 4 ) θ θ + π 2 y = sin(θ + π 2 ) π 2 0 π 2 π 3π 2 The graph of y = sin(θ π ) is shifted horizontally 4 Sometimes it is said that y = sin(θ π ) y = sinθ by 4 The graph of y = sin(θ + π ) is shifted horizontally 2 Sometimes it is said that y = sin(θ + π ) y = sinθ by 2
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 21 Example: Plot y = 2sin(3(θ π 2 )) 1!! Amplitude Vertical Shift So the graph is oscillating about Range!!! Phase Shift Period k=!! So the period is Think about the best starting place. That will be when the argument of sine function is zero. In this case, start at θ =, at that point y = The sine function returns to zero after half of period. The period is so half of that is. If we start at π 2 and go forward π 3. We ll be at. So at θ =, y =. Then again at another π 3.! 5π 6 + π 3 = Halfway between the zeros are the maximums and minimums of the function.
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 22 Plot y = cosθ and y = secθ on the same graph Remember, secθ is undefined when cosθ = 0 Plot y = tanθ (Talk about that one problem on WebAssign!)
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 23 Suggested Problems: Text: 1-11 My Previous Exams:!! S14 3A: 5!! F13 3A: 5!!!!! S13 3A: 6, 13! F12 3A: 10 Dr. Scarborough s Previous Exams:! F13 III p4:8, p8:10!!!!!!! F12 III p2: 2, p7: 4 Dr. Scarborough s Fall 2013 WIR 9: 26, 37 Dr. Kim s Fall 2014 WIR:
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 24 Chapter 8D - Trigonometric Identities In general, cos( θ) =!!! sin( θ) =!!!! tan( θ) =!!! sec( θ) =!!! csc( θ) =!!!! cot( θ) = Remember a function is even if, and a function is odd if So are even functions, and are odd functions.
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 25 Pythagorean Identities π π π π sin 2 + cos 2!!!!!! sin 2 + cos 2 3 3 4 4 sin 2 ( π) + cos 2 π ( )!!!!!! sin 2 ( θ) + cos 2 ( θ) sin 2 ( θ) + cos 2 θ ( )!!!!!! sin 2 ( θ) + cos 2 ( θ) Sum of Two Angles Formulas sin(α + β) cos(α + β) Difference of Two Angles Formulas sin(α β) cos(α β)
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 26 Sometimes these formulas are written like this: sin(α ± β) cos(α ± β) Double Angle Formula sin(2α) cos(2α) Getting from Double Angle Formulas to Square Formulas Square Formulas sin 2 α!!!!!!!! cos 2 α
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 27 The other one: Complementary Angle Identities sin π 2 θ =!!!!!!!! cos π 2 θ =!!!!!!!! Supplementary Angle Identities sin ( π θ) =!!!!!!!! cos( π θ) =!!!!!!!!
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 28 Using the Trigonometic Identities: 1. Determine the exact value of cos( 75 ) 2. If tanθ = x, express sin( 2θ ) in terms of x.
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 29 3. Given that tanα = 1 3 with α in Quadrant III and that cot β = 3 2 with β in Quadrant II, determine the exact values of the following: a) cscα!!!! b) sin2β!!!! c) cos( α β )
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 30 4. Simplify the expression secθ tanθ tanθ secθ cotθ
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 31 csc x 5. Verify tan x sin x = cot x + cot 3 x Suggested Problems: Text: 1, 3, 4, 5, 7, 9-13 My Previous Exams:! S14 3A: 7,! F13 3A: 1 a-f,!! F12 3A: 1, 12 Dr. Scarborough s Previous Exams:! F13 III p8:9,! F12III p2: 1, p7: 5 Dr. Scarborough s Fall 2013 WIR 9: 6, 49 Dr. Kim s Fall 2014 WIR:
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 32 Chapter 8E -- Inverse Sine Function The graph of y = sin x is beautiful, but clearly not 1-1. (Remember a function is 1-1 if ) But similar to other cases we studied earlier, the domain could be restricted, so that y = sin x is 1-1 on the restricted domain. Consider y = sin x for - π 2 x π 2. Is y = sin x a 1-1 function on this restricted domain? Reflect the curve drawn above through the line y=x. This new curve is the graph of Domain Range y = sin x y = arcsin x = sin 1 x x = arcsin y = sin 1 y!! iff
Note for - π 2 x π 2 and 1 y 1 sin 1 (sin x) = and sin(sin 1 y) = 1. sin 1 3 2!!!!!! 2. sin 1 2 2 3. sin 1 sin π 6!!!!!!! 4. sin sin 1 2 3 5 sin 1 sin 2π 3!!!!!! 6. cos sin 1 2 3 Suggested Problems: Text: 1-8 Dr. Scarborough s Fall 2013 WIR 10: 1, 2, 15 Dr. Kim s Fall 2014 WIR: Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 33
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 34 Chapter 8F - Inverse Trigonometric Functions Like y = sin x, y = cos x is not a 1-1 function unless the domain is restricted. For y = cos x, we will restrict the domain to 0 x π Consider the plot y = cos x for 0 x π. Reflect y = cos x through the line y = x This reflected curve is the curve of y = cos x y = arccos x = cos 1 x Domain Range!! x = arccos y =cos 1 y iff Note for 0 x π and 1 y 1 cos 1 (cos x) =! and cos(cos 1 y) =
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 35 1. Evaluate a) cos 1 ( 1) = 2 b) cos 1 = 2 c) cos 1 1 = 3 d) cos 1 = 2 2 e) cos cos 1 1 2 = f) cos 1 cos 5π 4 = Inverse Tangent Function Recall the graph of y = tan x To define an inverse function we limit the domain to - π 2 < x < π 2
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 36 Examine the plot of y = tan x for - π 2 < x < π 2 Is y = tan x a 1-1 function on this restricted domain?! Reflect y = tan x through the line y = x. This new curve is the graph of Domain Range y = tan x y = arctan x =tan 1 x!! x = arctan y =tan 1 y!! iff Note for - π 2 < x < π 2 and < y < tan 1 (tan x) = and tan(tan 1 y) = 2. Evaluate a) arctan( 1)!!!!!! b) arctan 3 3
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 37 c) sin arctan 4 5!!!!!! d) cos(arctan2) e) cot sin 1 12 13!!!!! f) tan(arccos( 3)) g) cos(arctan( 3)) Suggested Problems: Text: 1-8 Dr. Scarborough s Fall 2013 WIR 10: 4, 5, 7, 8, 10, 12, 13, 19 Dr. Kim s Fall 2014 WIR:
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 38 Chapter 8G - Law of Sines and Law of Cosines Given a general triangle, labeled as below Two interesting truths exist: A. The Law of Sines!!! sin A a = sin B b = sinc c B. The Law of Cosines:!!! c 2 = a 2 + b 2 2abcosC Notice that the Pythagorean Theorem is a special case of the Law of Cosines! In a right triangle, C = so cosc = and we have
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 39 Understanding the possibilities! In geometry, we learned that if 2 triangles have corresponding sides of the same length, then they are congruent. It is usually referred to as SSS Congruence Theorem. So if we are given a set of 3 numbers that represent the lengths of the sides of a triangle, then there are two possibilities: either there is no triangle with those side lengths or there is exactly 1 triangle with those side lengths. 1. If a = 3, b = 4, and c = 8, we know that no triangle exists because 3 + 4 < 8 2. If a = 2, b = 3, and c = 4, then we could use the Law of Cosines to determine the measures of the 3 angles.
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 40 There was the SAS Congruence Theorem. It told us that if 2 sides and the included angle of one triangle are congruent to the corresponding sides of another triangle, then the 2 triangles are congruent. From this we know that if we are given the lengths of 2 sides of a triangle and the measure of the included angle, then there is exactly one triangle with these measurements. We can use the Law of Cosines to calculate the remaining measurements. 3. If a=14, b=16, and C = 120, determine the length of side c. Here, too, we would need to start with the Law of Cosines. There were also the AAS and ASA Congruence Theorems. They told us that if 2 angles and one of the sides of a triangle was congruent to the corresponding parts of another triangle, then the triangles were congruent. This could be interpreted another way. If we are given 2 angle measures whose sum is less than 180 and the length of one of the sides of the triangle, then we can determine the angle measure of the third angle and the length of the other 2 sides of the triangle. Here the Law of Sines is the easiest to use. 4. If A = 75, C= 65, and a=3, solve the triangle.
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 41 Finally we get to the congruence theorem that does not exist: SSA. If you have 2 triangles, and you know that 2 sides on one triangle are congruent to 2 sides on the other triangle and one of the angles from one triangle is congruent to an angle on the other triangle, but it s not the angle between the congruent sides, then you cannot be sure that the triangles are congruent. For us this means that if we are given 2 numbers that represent the lengths of the sides of a triangle and an angle measure that is not between the given sides, then there are three possibilities for that set of numbers. 1. The given conditions might be such that no triangle exists. 2. The given conditions might be such that exactly 1 triangle exists. 3. The given conditions might be such that 2 different triangle exist. Example: Triangle ABC does not necessarily exist. If it does, determine the missing parts of the triangle from the given information. 5. A = 75, a = 51, b = 71
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 42 6..A = 37, a = 12, b = 16.1
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 43 7. A = 40, a = 20, b = 15
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 44 8. From (http://cims.nyu.edu/~kiryl/precalculus/section_6.5-the%20law%20of%20sines/the %20Law%20of%20Sines.pdf) a)
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 45 b)
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 46 c)
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 47 9. (From http://www.math.usu.edu/~rheal/online1060/chapter7/7-2.pdf)
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 48 10. (From http://www.math.usu.edu/~rheal/online1060/chapter7/7-3.pdf ) Suggested Problems: Text:4-11!!! Dr. Scarborough s Fall 2013 WIR 10: 3, 6, 9, 14, 17 Dr. Kim s Fall 2014 WIR:
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 49 Chapter 8H - Solving Trigonometric Equations For what values of θ does sin 2 θ + cos 2 θ = 1? sin 2 θ + cos 2 θ = 1 is true for because sin 2 θ + cos 2 θ = 1 is a trigonometric. Solving Trigonometric Equations: 1. Solve sinθ = 3 2
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 50 2. Solve sin 2 ( θ ) = 3 4
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 51 3. a) Solve cos( 3θ ) = 1 b) Solve cos( 3θ ) = 1, 0 θ 2π
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 52 4. a) Solve ( sin x) ( tan x) = sin x
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 53 b) Which of these solutions is on [ π, π] 5. Solve cos x = sin2x on [ π, π].
6. Solve 2 sin x 2 2 + 3sin x 2 +1 = 0 on [ 4π, 4π ]. Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 54
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 55 7. Consider the identity sin2x = 2sin x cos x. 1 It could be rewritten as 2 sin2x = sin x cos x. a) What is the amplitude of the function f (x) = 1 2 sin2x? b) What is the period?!! c) Plot f (x) = 1 2 sin2x d) For what values of θ does (sin 2 θ)(cos 2 θ) = 1?
Fry Texas A&M University!! Math 150!! Chapter 8!! Fall 2014! 56 e) For what values of θ on [ π, π] does 4(sin 2 θ)(cos 2 θ) = 1? Suggested Problems: Text: 1-18!!!! Dr. Scarborough s Fall 2013 WIR 10: 11, 16, 18 Dr. Kim s Fall 2014 WIR: