Habeman MTH Section III: ola Coodinates and Comple Numbes Chapte : Intoduction to ola Coodinates We ae all comfotable using ectangula (i.e., Catesian coodinates to descibe points on the plane. Fo eample, we ve plotted the point = (, on the coodinate plane in Figue. ' Figue : The point = (, on the ectangula coodinate plane. Instead of using these ectangula coodinates, we can use a cicula coodinate sstem to descibe points on the plane: ola Coodinates. Odeed pais in pola coodinates have fom (, θ whee epesents the point s distance fom the oigin and θ epesents the angula displacement of the point with espect to the positive -ais. Let s find the pola coodinates that descibe in Figue. Fist let s find, the distance fom point to the oigin; in othe wods, we need to find the length of the segment labeled in Figue : Figue ' Figue We can use the thagoean Theoem to find :
Habeman MTH Section III: Chapte = + = = ( ( = + Now we need to find the angle between the positive -ais and the segment labeled ; this angle is labeled θ in Figue. = θ ' Figue We can use the ight tiangle induced b the angle θ and the side along with eithe sine o cosine to find the value of θ : sin( θ = θ = Thus, in pola coodinates, (, plane in Figue. =. We ve plotted the point on the pola coodinate = ' Figue : The point (, = on the pola coodinate l
Habeman MTH Section III: Chapte EXAMLE : lot the point A = on the pola coodinate plane and detemine the ectangula coodinates of point A. SOLUTION: To plot the point A (, θ, so A = we need to ecognize that pola odeed pais have fom = implies that We ve plotted the point A = 0 and θ =. = on the pola coodinate plane in Figue. θ = = 0 A Figue To find the ectangula coodinates of point A we can use the efeence angle fo θ, which is, and the induced ight tiangle; see Figue. A 0 Figue
Habeman MTH Section III: Chapte Using the tiangle in Figue, we can see that cos ( = 0 = 0cos = 0 = ( and sin ( = 0 = 0sin = 0 = ( Since point A is in Quadant III, we know that both and ae negative. Thus, the ectangula coodinates of point A ae (,. EXAMLE : Find the ectangula coodinates of a geneic point (, θ coodinate plane. = on the pola SOLUTION: In Figue 7, we've pointed plotted in the pola plane. θ = (, θ Figue 7 We can constuct a ight tiangle and use tigonomet to obtain epessions fo the hoizontal and vetical coodinates of point ; see Figue 8 below.
Habeman MTH Section III: Chapte θ Figue 8 Based on the tiangle in Figue 8, we can see that cos( θ = = cos( θ and sin( θ = = sin( θ = is epesents a point on the pola coodinate plane, then the =. (Notice that we obseved Thus, if (, θ ectangula coodinates of ae (, ( cos( θ, sin( θ essentiall the same fact in Section I: Chapte. We can use what we ve discoveed to tanslate pola coodinates into ectangula coodinates. The pola coodinates (, θ ae equivalent to the ectangula coodinates ( θ θ (, = cos(, sin(. Ke oint: ola and ectangula odeed pais cannot be set equal to each othe. When odeed pais ae descibed as being equal, it means that the have 0.7, 0. =, the same coodinates so we can wite something like ( ( 0.7 = and 0. = but we can t wite = (, since (fom Eample since 0 and. In ode to communicate that ectangula odeed pais and pola odeed pais descibe the same location, we need to compose sentences like, The ectangula 0,. odeed pai (, is equivalent to the pola odeed pai (
Habeman MTH Section III: Chapte EXAMLE : lot the point B (, SOLUTION: ectangula coodinates of the point. To plot the point B (, (, θ, so B (, = on the pola coodinate plane and find the = we need to ecognize that pola odeed pais have fom = implies that = and θ =. Hee, is negative. This means that when we get to the teminal side of θ =, instead of going fowad units into Quadant II, we need to go backwads units into Quadant IV; see Figue 9. θ = = B Figue 9 To find the ectangula coodinates of point B, we can use the convesion equations we deived in the pevious eample. = cos( θ = cos = = ( ( and = sin( θ = sin = = ( ( Thus, the ectangula coodinates of B ae (, = (,.