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1.1 -- Angles Wednesday, August 07, 2013 12:53 PM Basic Terminology: Line AB Segment AB Ray AB Angles in Standard Position Animation of terminal side of an angle Sketch a picture of the following angle measures. 1. 240⁰ 2. 50⁰ Naming an Angle: • It can be named by its vertex or • by using 3 points, making sure the vertex is the middle letter. 4. 3. 150⁰ B• A Degree Measure: We assign 360 degrees to a full rotation of a ray. Chapter 1 Page 1 • C 300⁰ The Greek letter Ө (theta) is used to name each angle. Other Greek letters such as (alpha) and β (beta) are also used often. Acute angle Right Angle Obtuse Angle Complementary Angles Straight Angle Supplementary Angles Ex 1: Finding the complement and the Supplement of an Angle Classroom Example 1: For an angle measuring 40˚ For an angle measuring 55˚ a) Find the complement a) Find the complement b) Find the supplement b) Find the supplement Ex2: Find the measure of each angle in the give figures. 2) 1) 4x˚ 6x˚ 6x˚ 3x˚ 3) 4) 2x˚ 7x˚ 3x˚ Chapter 1 Page 2 2x˚ The measure of an angle can be written where it is read "the measure of angle A = 35" Traditionally, portions o a degree have been measured with minutes and seconds. One minute is written 1'. One second is written 1". or or 12˚ 42' 38" = 12 degrees, 42 minutes 38 seconds Ex3: Calculating with Degrees, Minutes, and Seconds Perform each calculation: 1) 51˚ 29' + 32˚ 46' 2) 90˚ Classroom Ex3: Perform each calculation. 1) 28˚ 35' + 63˚ 52' 2) 180 73˚ 12' 117˚ 29' Because calculators are so prevalent, angles are commonly measured in decimal degrees. We will be converting them to degrees, minutes, and seconds (D˚ M' S"). For example 12.4238˚ (your calculator may be able to make those conversions for you. Ex4: Converting between Decimal Degrees and D˚ M' S" 1) Convert 74˚ 08' 14" 2) convert 34.817˚ to decimal degrees to the nearest thousandth. to D˚ M' S" to the nearest second Chapter 1 Page 3 Classroom Ex4: 1) Convert 105˚ 20' 32" 2) convert 85.263˚ to decimal degrees to the nearest thousandth. to D˚ M' S" to the nearest second Quad Angle Animation Quadrant Angles 's in standard position whose terminal sides lie on the x-axis or the y-axis such as angles with measures 90˚, 180˚, and 270˚ are Quadrant Angles. Coterminal Angles Coterminal Angle Animation Angles that continue past 360˚ are coterminal angles. Angles with measures of 60˚ and 420˚ have the same initial side and terminal side, but they differ in the amount of rotation. Their measures differ by multiples of 360˚. Ex5: Finding Measures of Coterminal Angles Find the angles of least positive measure that are coterminal with each angle. 1) 908˚ 2) 75˚ 3) 800˚ 1) 1106˚ 5) 150˚ 6) 603˚ Ex6: Analyzing the Revolutions of a CD player CD players always spin at the same speed. Suppose a CD player makes 480 revolutions per min. Through how many degrees will a point on the edge of a CD move in 2 sec? Chapter 1 Page 4 Classroom Ex6: A wheel made 270 revolutions per min. Through how many degrees will a point on the edge of the wheel move in 5 seconds? Chapter 1 Page 5 1.2 -- Angle Relationships and Similar Triangles Wednesday, August 07, 2013 12:55 PM Geometric Properties • Vertical Angles are congruent Parallel Lines form angles that are congruent or supplementary 1) The corr are 2) The alt int 3) The alt ext are are 4) The cons int are supp Ex1: Finding Angle Measures Find the measures of angles 1, 2, 3, and 4 given that the lines are parallel. 1 (3x + 2)˚ 1 2 2 3 4 (9x + 9)˚ 3 (7x - 5)˚ (5x + 40)˚ 4 Triangles • The angle sum of a triangle is 180 Ex2: Applying the Angle Sum Triangle Property 1. Find the measure of the 3rd angle of a triangle that has 2 angles with measures of 48˚ and 61˚. 2. Find the measure of the 3rd angle of a triangle that has 2 angles with measures of 33˚ and 26˚. 3. One angle is 36⁰ 43' and the 2nd angle is 58⁰ 29', find the measure of the 3rd angle. Chapter 1 Page 6 ANGLES SIDES Types of triangles. Similar Triangles: Triangles that are similar have exactly the same shape but not necessarily the same size. Similar Similar Similar and Congruent Conditions for Similar Triangles 1) All the corresponding angles must be congruent 2) All of the corresponding sides must be proportional (that is, the ratios of the corresp sides must be equal) Ex3: Finding angle measures in similar triangles Find the measure of all angles, given that ΔABC is similar to ΔDEF 104˚ 45˚ Chapter 1 Page 7 Ex4: Finding Side Lengths in Similar Δs 24 16 The triangles are similar, find the length of each side. 8 32 12 y 12 8 10 x The exterior angle of a triangle is equal to the sum of the 2 remote interior angles of the triangle. 1 2 4 3 Chapter 1 Page 8 Because if you add angle 4 to each side the total will be 180⁰. Therefore they are equal Quiz Review? Thursday, August 08, 2013 2:40 PM Chapter 1 Page 9 1.3 -- Trig Functions Wednesday, August 07, 2013 12:56 PM Trigonometric Functions Ex1: Finding Function Values of an Angle The terminal side of an angle θ in standard position passes through the point (8, 15). Find the 6 trig functions of angle θ. sin θ = csc θ = cos θ = sec θ = tan θ = cot θ = Classroom Ex1: The terminal side of an angle θ in standard position passes through the point (12, 5). Find the 6 trig functions of angle θ. Ex2: Finding Function Values of an Angle Chapter 1 Page 10 Ex2: Finding Function Values of an Angle The terminal side of an angle θ in standard position passes through the point ( 3, 4). Find the values of the 6 trig functions of angle θ. x= 3 y= 4 r= θ -3 -4 (-3, -4) Classroom Ex2:The terminal side of an angle θ in standard position passes through the point (8, 6). Find the values of the 6 trig functions of angle θ. ** We can also find the trig function values of an angle if we know the equation of the line coinciding with the terminal ray. Recall from algebra that the graph of the equation is the line that passes through the origin. (in standard form) Ex3: Finding Function Values of an Angle Find the 6 trig functions of the angle θ in standard position, if the terminal side of θ is defined by x + 2y = 0, and x ≥ 0. (1) Pick any value of x (the easiest one is the coefficient of y - because then you won't get fraction answers) then (2) plug it into the equation and solve for y. (3) Use that (x, y) point to solve for r. Classroom Ex3:Find the 6 trig functions of the angle θ in standard position, if the terminal Chapter 1 Page 11 Classroom Ex3:Find the 6 trig functions of the angle θ in standard position, if the terminal side of θ is defined by 3x 2y = 0, and x ≤ 0. Slope-intercept form of a line: y = mx + b In Ex3, the equation x + 2y = 0 can be written as, So the slope is . Notice that In general, it is true that The reciprocal of 0 is undefined. Quadrantal Angles Remember that quadrantal angles 0˚, 90˚, 180˚, 270˚, and 360˚. When determining trig function values of quadrantal angles, the figure below can help find the ratios. Because any point on the terminal side can be used, it is convenient to choose the point 1 unit from the origin, with r = 1. (in ch3 this idea will extend to the unit circle.) (0, 1) x = -1 y=0 r=1 (-1, 0) x=0 y=1 r=1 90˚ x=0 y = -1 r=1 (1, 0) x=1 y=0 r=1 sin 90 = csc 90 = cos 90 = sec 90 = tan 90 = cot 90 = (0, -1) Ex4: Find the values of the 6 trig functions for an angle θ in standard position with terminal side through ( 3,0) Chapter 1 Page 12 Use the trig function values of quadrantal angles to evaluate each expression. If you see cot2 90⁰ it means (cot 90⁰)2 Chapter 1 Page 13 1.4 -- Using the Definitions of the Trig Functions Wednesday, August 07, 2013 12:56 PM Identities are equations that are true for all values of the variables for which all expressions are defined. Reciprocal identities: For all angles θ for which both functions are defined, the following identities hold. CAUTION Be sure not to use the inverse trig keys on your calculator to find reciprocals. For ex: Ex1: Using the Reciprocal Identities 1) Find cos θ, given that sec θ = 2) Find sin θ, given that csc θ = 3) Find tan θ, given that cot θ = 4 4) Find sec θ, given that cos θ = Sign and Ranges of Function Values All Students Take Calculus Q1 -- All functions are positive QII -- Sin and csc are positive QIII -- Tan and cot are positive QIV -- Cos and sec are positive Ex2: Determining Signs of Functions of Nonquadrant Angles Chapter 1 Page 14 Ex2: Determining Signs of Functions of Nonquadrant Angles Determine the quadrant of the given angle and name which trig functions are positive in that quadrant. 1. 87˚ 2) 300˚ 3) 200˚ 1. 54˚ 5) 260˚ 6) 60˚ Ex3: Identifying the Quadrant of an Angle Identify the quadrant (or possible quadrants) of an angle θ that satisfies the given conditions. 1) 2) 4) 3) -1 ≤ sin θ ≤ 1 -1 ≤ cos θ ≤ 1 sec θ ≤ -1 OR sec θ ≥ 1 csc θ ≤ -1 OR csc θ ≥ 1 tan θ can be any real # cot θ can be any real # Ex4: Deciding Whether a Value is in the Range of a Trig Function Decide whether each statement is possible or impossible 1) sin θ = 2.5 2) cot θ = 0.999 Chapter 1 Page 15 2) tan θ = 110.47 3) sec θ = 0.6 5) cos θ = 1.7 6) csc θ = 0 2) cot θ = 0.999 5) cos θ = 1.7 6) csc θ = 0 Ex5: Finding All Function Values Given One Value and the Quadrant 1) Suppose that angle θ is in Q II and sin θ = . Find the values of the other 5 trig functions. Classroom Ex5: Suppose that angle θ is in Q III and tan θ = . Find the values of the other 5 trig functions. Pythagorean Identities From x2 + y2 = r2 we can derive 3 new identities. As before, we are only given one form of each identity. However, with algebra we can rewrite each of these. For example. is equivalent to ** it is important to be able to transform these identities quickly and to recognize their equivalent forms! Chapter 1 Page 16 Quotient Identities: So let's look at the quotient of ** note: the denominators cannot = 0 Ex6: Using Identities to Find Function Values Find sin θ and tan θ, given that and sin θ > 0 Classroom Ex6: Find cos θ and tan θ, given that sin θ = and cos θ > 0. Ex7: Using Identities to Find Function Values ** we can use x2 + y2 = r2 Find sin θ and cos θ, given that tan θ = and θ is in QIII. Using Pythagorean and quotient identities OR x 2 + y 2 = r2 Classroom Ex7: Find sin θ and cos θ, given that cot θ = Chapter 1 Page 17 and θ is in QII. Classroom Ex7: Find sin θ and cos θ, given that cot θ = Chapter 1 Page 18 and θ is in QII. Test Review? Thursday, August 08, 2013 2:40 PM Chapter 1 Page 19