Fry Texas A&M University Math 150 Chapter 8A Fall 2015! 207 Since 1 revolution = 1 = = Since 1 revolution = 1 = = Convert to revolutions (or back to degrees and/or radians) a) 45! = b) 120! = c) 450! = d) 3 1 3 revolutions = e) 5π 4 radians = f) 11π 3 radians = Sketch θ 1 = 3π and θ 2 = π on the same unit circle 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
Fry Texas A&M University Math 150 Chapter 8A Fall 2015! 208 List 2 other angles that are coterminal angles with π 2 List 2 other angles that are coterminal angles with 2π 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 8A Fall 2015! 209 Arc Length: (Think about the fraction of the circumference.) θ = 120! r = 6 units θ = π 4 r = 5 units Area of a sector: (Think about the fraction of the area of the circle.) θ = 30!, r = 2 units θ = 7π 6 r = 3 units
Fry Texas A&M University Math 150 Chapter 8A Fall 2015! 210 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 =! Extra Problems:! Text: 1-12 1. A given circle has radius 7 meters. Determine the area of a sector of this circle with central π angle measuring 5 radians. 2. If a circle has a radius of 10 centimeters, find the exact area of the sector subtended by a central angle of 11π 25 radians.
Fry Texas A&M University Math 150 Chapter 8B Fall 2015! 219 1. Given that sinθ = 3 5 and θ is acute, determine the values of a) cosθ = b) tanθ = 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 8B Fall 2015! 220 4. Given that secθ = 7 5 and tanθ < 0 determine the value of tanθ. Extra Problems: Text: 3-11 1. A 12 foot ladder is resting against a vertical wall. If the angle between the top of the ladder and the wall is 30, what is the distance from the base of the ladder to the wall? 2. If tanθ = 3 3 and sinθ < 0 is then cosθ = (Give your answer in simplified form.)
Fry Texas A&M University Math 150 Chapter 8C Fall 2015! 221 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 8C Fall 2015! 222 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 8C Fall 2015! 223 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 8C Fall 2015! 224 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 8C Fall 2015! 225 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 8C Fall 2015! 226 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 8C Fall 2015! 227 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 8C Fall 2015! 228 Extra Problems:! Text: 1-11 1. Label each of the graphs: b) c) d) e) 2. Given f (x) = 4 cos(3x) + 2 a) What is the amplitude of this function? b) What is the period of this function? c) State the range of this function in interval notation. π d) f 9 =
Fry Texas A&M University Math 150 Chapter 8C Fall 2015! 229 3. Given h(x) = 2sin( 4x) 1then provide the following. (If there are none, write NONE in the blank provided.) a) domain (in interval notation): b) range (in interval notation): c) coordinates of the y-intercept d) period e) graph h(x) = 2sin( 4x) 1on [0, 2π ] f) coordinates of the x-intercept(s) on [0, 2π ]
Fry Texas A&M University Math 150 Chapter 8D Fall 2015! 230 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 hence symmetric While are odd functions and so are symmetric
Fry Texas A&M University Math 150 Chapter 8D Fall 2015! 231 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 8D Fall 2015! 232 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 8D Fall 2015! 233 The other one: Complementary Angle Identities Remember if two angles are complementary, then their sum is. If θ 1 = π 6, its complement is θ 2 = If θ 1 = π 4, its complement is θ 2 = If we have a general angle θ its complement is sin π 2 θ =!!!!!!!! cos π 2 θ =!!!!!!!! In words, this is saying that the sine of the complement of an angle is equal to the and the cosine of the complement of an angle is equal to the
Fry Texas A&M University Math 150 Chapter 8D Fall 2015! 234 Supplementary Angle Identities Remember if two angles are supplementary, then their sum is. If θ 1 = π 3, its supplement is θ 2 = If θ 1 = 5π 6, its supplement is θ 2 = If we have a general angle θ its complement is sin ( π θ) =!!!!!!!! cos( π θ) =!!!!!!!! In words, this is saying that the sine of the supplement of an angle is equal to the and the cosine of the supplement of an angle is equal to the Using the Trigonometric Identities: 1. Determine the exact value of cos( 75 )
Fry Texas A&M University Math 150 Chapter 8D Fall 2015! 235 2. 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 8D Fall 2015! 236 3. If tanθ = x, express sin( 2θ ) in terms of x. 4. Verify tan x sin x = csc x cot x + cot 3 x
Fry Texas A&M University Math 150 Chapter 8D Fall 2015! 237 5. Simplify the expression secθ tanθ tanθ secθ cotθ Extra Problems: Text: 1, 3, 4, 5, 7, 9-13 1. ( sin x + cos x) 2 = a) 1+ cos2x b) 1+ sin2x c) tan 2 x d) 1 None of these
Fry Texas A&M University Math 150 Chapter 8D Fall 2015! 238 2. Simplify cos x + tan xsin x a) sec x b) sin x c) cos x d) tan x e) None of these 3. Exactly evaluate cos θ + π where cosθ = 2 6 5 and tanθ < 0.