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6/4/13 Obj: SWBAT plot polar coordinates Bell Ringer: Plot the point (4, 4π/3) HW: Polar Coordinates WS Announcements: • Turn in missing Assignments (Excused) • Take missing Quizzes • Last day to bring books back is day of the Final • Turn in 3 missing assignments partial credit; by Friday • Check grades to look for missing assignments (r, ) You are familiar with plotting with a rectangular coordinate system. We are going to look at a new coordinate system called the polar coordinate system. The center of the graph is called the pole. Angles are measured from the positive x axis. Points are represented by a radius and an angle radius (r, ) To plot the point 5, 4 First find the angle Then move out along the terminal side 5 A negative angle would be measured clockwise like usual. 3 3, 4 To plot a point with a negative radius, find the terminal side of the angle but then measure from the pole in the negative direction of the terminal side. 2 4, 3 Let's plot the following points: 7, 2 7 , 2 3 5 7, 7 , 2 2 Notice unlike in the rectangular coordinate system, there are many ways to list the same point. Let's take a point in the rectangular coordinate system and convert it to the polar coordinate system. (3, 4) r 4 3 Based on the trig you know can you see how to find r and ? 3 4 r 2 2 2 r=5 tan 3 We'll find in radians polar coordinates are: (5, 0.93) tan 4 1 4 0 . 93 3 Let's generalize this to find formulas for converting from rectangular to polar coordinates. (x, y) r y x y r 2 r 2 2 x y 2 2 x tan y x You need to consider the quadrant in which P lies in order to find the value of . tan 1 y x Now let's go the other way, from polar to rectangular coordinates. Based on the trig you know can you see how to find x and y? 4, 4 4 y cos 4 4 x x 4 2 2 2 x 4 2 sin rectangular coordinates are: 2 2 , 2 2 4 y 4 2 2 2 y 4 2 Let's generalize the conversion from polar to rectangular coordinates. cos r , r y x x r x r cos sin y r y r sin Polar coordinates can also be given with the angle in degrees. 120 (8, 210°) 90 60 45 135 30 150 (6, -120°) 180 0 330 210 315 225 (-5, 300°) 300 240 270 (-3, 540°) To find the rectangular coordinates for a point given its polar coordinates, we can use the trig functions. Example 4, 3 12 13 Likewise, we can find the polar coordinates if we are given the rectangular coordinates using the trig functions. Think about the Pythagorean Theorem. Example: Find the polar coordinates (r, θ) for the point for which the rectangular coordinates are (5, 4). Express r and θ (in radians) to three sig digits. (5, 4) 14 Conversion from Rectangular Coordinates to Polar Coordinates If P is a point with rectangular coordinates (x, y), the polar coordinates (r, ) of P are given by r x y 2 2 ref tan 1 y x You need to consider the quadrant in which P lies in order to find the value of . P 15 Find polar coordinates of a point whose rectangular coordinates are given. Give exact answers with θ in degrees. a) 2, 2 16 Find polar coordinates of a point whose rectangular coordinates are given. Give exact answers with θ in degrees. b) 1, 3 17 The TI-84 calculator has handy conversion features built-in. Check out the ANGLE menu. 5: Returns value of r given rectangular coordinates (x, y) 6: Returns value of given rectangular coordinates (x, y) 7: Returns value of x given polar coordinates (r, ) 8: Returns value of y given polar coordinates (r, ) Check the MODE for the appropriate setting for angle measure (degrees vs. radians). 18 Convert the rectangular coordinate system equation to a polar coordinate system equation. From conversion s, how 2 2 x y 9 2 2 r x y r 3 2 2 was r related to x and y ? Here each r unit is 1/2 and we went out 3 and did all angles. Before we do the conversion let's look at the graph. r must be 3 but there is no restriction on so consider all values. Convert the rectangular coordinate system equation to a polar coordinate system equation. 2 x 4y What are the polar conversions we found for x and y? substitute in for x and y x r cos r cos y r sin 2 4 r sin r cos 4 r sin 2 2 We wouldn't recognize what this equation looked like in polar coordinates but looking at the rectangular equation we'd know it was a parabola. Write Polar Equation in Rectangular Form Given r = 2 sin θ x r co s – Write as rectangular equation y r sin y Use definitions – And identities tan 1 x r x y 2 2 2 – Graph the given equation for clues 21 Write Polar Equation in Rectangular Form Given r = 2 sin θ – We know – Thus – And r sin 2 y r r 2y 2 x y 2y 2 2 22 Write Rectangular Equation in Polar Form Consider 2x – 3y = 6 x r co s y r sin – As before, use definitions y tan 1 x 2 r cos 3 r sin 6 r x y 2 2 2 r 2 cos 3 sin 6 r 6 2 cos 3 sin 23 Polar Coordinates How do we graph or use a graph to describe loops and curves? Background The use of polar coordinates allows for the analysis of families of curves difficult to handle through rectangular coordinates (x,y). If a curve is a rectangular coordinate graph of a function, it cannot have any loops since, for a given value there can be at most one corresponding value. However, using polar coordinates, curves with loops can appear as graphs of functions. http://www.xpmath.com/careers/topicsresult.php ?subjectID=4&topicID=18 24 25 Polar Coordinates (Sect 21.10) A point P in the polar coordinate system is represented by an ordered pair r , . • If r 0 , then r is the distance of the point from the pole. • is an angle (in degrees or radians) formed by the polar axis and a ray from the pole through the point. 26 Example Plot the point P with polar coordinates 2, 4 . 27 Polar Coordinates If r 0 , then the point is located r units on the ray that extends in the opposite direction of the terminal side of . For r 0 , r = |r| and ϴ = ϴ + 180 Ex. (-7, 70⁰) = (7, 250) 28 Example Plot the point with polar coordinates 4, 3 29 Plotting Points Using Polar Coordinates a) 5 3, 3 b) 2, 4 30 Plotting Points Using Polar Coordinates c) 3, 0 d) 5, 2 31 A) B) C) D) 32 • Using polar coordinates, the same point can be described by many different representations. Which of the following do(does) not describe the point (8,60°) ? (8,−60°) (8,-300°) (8,420°) (-8,-120°) • Using polar coordinates, write 3 more representations of the point (5,150°) 33 Graphing in polar coordinates • Hit the MODE key. • Arrow down to where it says Func (short for "function" which is a bit misleading since they are all functions). • Now, use the right arrow to choose Pol. • Hit ENTER. (*It's easy to forget this step, but it's crucial: until you hit ENTER you have not actually selected Pol, even though it looks like you have!) The calculator is now in polar coordinates mode. To see what that means, try this. Hit the Y= key. Note that, instead of Y1=, Y2=, and so on, you now have r1= and so on. In the r1= slot, type 5-5sin( Now hit the familiar X,T,q,n key, and you get an unfamiliar result. In polar coordinates mode, this key gives you a ϴ instead of an X. Finally, close off the parentheses and hit GRAPH. If you did everything right, you just asked the calculator to graph the polar equation r=5-5sin(ϴ). The result looks a bit like a valentine. The WINDOW options are a little different in this mode too. You can still specify X and Y ranges, which define the viewing screen. But you can also specify the ϴvalues that the calculator begins and ends with; for instance, you may limit the graph to 0< ϴ <π/2. This would not change the viewing window, but it would only draw part of the graph. 34 End here 6/3 35 End of Section 36 Definitions of Trig Functions of Any Angle (Sect 8.1) Definitions of Trigonometric Functions of Any Angle Let be an angle in standard position with (x, y) a point on the terminal side of and r x 2 y 2 y sin cos tan y csc r r y x r (x, y) r x r y x x sec cot x y 37 The Signs of the Trig Functions Since the radius is always positive (r > 0), the signs of the trig functions are dependent upon the signs of x and y. Therefore, we can determine the sign of the functions by knowing the quadrant in which the terminal side of the angle lies. 38 The Signs of the Trig Functions 39 Where each trig function is POSITIVE: “All Students Take Calculus” Translation: S T A C A = All 3 functions are positive in Quad 1 S= Sine function is positive in Quad 2 T= Tangent function is positive in Quad 3 C= Cosine function is positive in Quad 4 *In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and tangent are negative. **Reciprocal functions have the same sign. So cosecant is positive wherever sine is positive, secant is positive wherever cosine is positive, and cotangent is positive 40 wherever tangent is positive. Example Determine if the following functions are positive or negative: sin 210° cos 320° cot (-135°) csc 500° tan 315° 41 Examples For the given values, determine the quadrant(s) in which the terminal side of θ lies. 1) sin 0.3614 2) tan 2.553 3) cos 0.866 42 Examples Determine the quadrant in which the terminal side of θ lies, subject to both given conditions. 1) sin 0, cos 0 2) sec 0, cot 0 43 Examples Find the exact value of the six trigonometric functions of θ if the terminal side of θ passes through point (3, -5). 44 Reference Angles (Sect 8.2) The values of the trig functions for non-acute angles (Quads II, III, IV) can be found using the values of the corresponding reference angles. Definition of Reference Angle Let be an angle in standard position. Its reference angle ref is the acute angle formed by the terminal side of and the horizontal axis. 45 Example Find the reference angle for 2 2 5 Solution y By sketching in standard position, we see that it is a 3rd quadrant angle. To find ref , you would subtract 180° from 225 °. ref 225 180 ref x ref 45 46 So what’s so great about reference angles? Well…to find the value of the trig function of any non-acute angle, we just need to find the trig function of the reference angle and then determine whether it is positive or negative, depending upon the quadrant in which the angle lies. For example, sin 225 (sin 45 ) 1 2 In Quad 3, sin is negative 45° is the ref angle 47 Example Give the exact value of the trig function (without using a calculator). cos 150 48 Examples (Text p 239 #6 & 8) Express the given trigonometric function in terms of the same function of a positive acute angle. 6) tan 91 , sec 345 8) cos 190 , cot 290 49 Now, of course you can simply use the calculator to find the value of the trig function of any angle and it will correctly return the answer with the correct sign. Remember: Make sure the Mode setting is set to the correct form of the angle: Radian or Degree To find the trig functions of the reciprocal functions (csc, sec, and cot), use the button or enter [original function] . 50 Example Evaluate co t 3 2 4 .0 . Round appropriately. Set Mode to Degree Enter: OR A N S : cot 324.0 1.38 51 HOWEVER, it is very important to know how to use the reference angle when we are using the inverse trig functions on the calculator to find the angle because the calculator may not directly give you the angle you want. y Example: Find the value of to the nearest 0.01° -12 x r -5 (-12, -5) 52 Examples F in d fo r 0 3 6 0 1) sin 0.418 2) tan 1.058 53 Examples F in d fo r 0 3 6 0 3) cos 0.85 4) cot 0.012, sin 0 54 BONUS PROBLEM Find for 0 360 w ithout using a calculato r. 2 sin 1 0 55 SUPER DUPER BONUS PROBLEM Find for 0 360 w ithout using a calculato r. 4(sin ) 9 6 2 56 Trig functions of Quadrantal Angles To find the sine, cosine, tangent, etc. of angles whose terminal side falls on one of the axes (..., 180 , 90 , 0 , 90 , 180 , 270 , 360 , ...) , we will use the circle. (0, 1) 90 Unit Circle: Center (0, 0) radius = 1 x2 + y2 = 1 (-1, 0) (1, 0) 180 0 270 (0, -1) 57 Now using the definitions of the trig functions with r = 1, we have: sin cos y y csc r 1 r 1 y y x x r 1 r tan y y x x sec 1 x cot x x y 58 Example Find the value of the six trig functions for 9 0 (0, 1) sin 90 270 (-1, 0) (1, 0) 180 90 (0, -1) 0 cos 90 tan 90 y y r 1 x x r y 1 x csc 90 r y y sec 90 r 1 cot 90 x x 1 x y 59 Example Find the value of the six trig functions for 0 sin 0 y co s 0 x tan 0 y x csc 0 1 sec 0 1 co t 0 x y x y 60 Example Find the value of the six trig functions for 540 sin 540 y cos 540 x tan 540 y x csc 540 1 sec 540 1 cot 540 x y x y 61 Radian Measure (Sect 8.3) A second way to measure angles is in radians. Definition of Radian: One radian is the measure of a central angle that intercepts arc s equal in length to the radius r of the circle. In general, for in radians, s r 62 Radian Measure 2 radians corresponds to 360 2 6.28 radians corresponds to 180 3.14 2 radians corresponds to 90 1.57 2 63 Radian Measure 64 Conversions Between Degrees and Radians 1. 2. To convert degrees to radians, multiply degrees by To convert radians to degrees, multiply radians by 180 180 Example Convert from degrees to radians: 210º 210 65 Conversions Between Degrees and Radians Example a) Convert from radians to degrees: 3 3 4 4 b) Convert from radians to degrees: 3.8 3 .8 66 Conversions Between Degrees and Radians c) Convert from degrees to radians (exact): 6 7 5 675 d) Convert from radians to degrees: 13 13 6 6 67 Conversions Between Degrees and Radians Again! e) Convert from degrees to radians (to 3 decimal places): 5 2 52 f) Convert from radians to degrees (to nearest tenth): 1 rad 1 68 Examples Find to 4 sig digits for 0 2 sin 0.9540 69