Math IV Mathematical Models Topic V Other concepts of the line Teacher: Javier Licurgo Pedraza El estudiante aprenderá lo siguiente: El ángulo entre dos líneas Familia de líneas de una ecuación Coordenadas Polares The student must work on the following: Exercises written in the Exercise 5 section of CANVAS Exercises written in the Exercise 5 section of CANVAS Angle between two lines Two distinct lines in a plane are either parallel or intersecting. If they intersect and are non-perpendicular, their intersection forms two pairs of opposite angles. Trigonometric functions are used to find the angle between two lines that lie in the same plane. we can find an expression to find the angle between two lines by using only information about their slopes 𝑚1 and 𝑚2 . The angle formed between two straight lines can be approximated by relating their slopes tan 𝜃 = 𝑚2 − 𝑚1 1 + 𝑚1 𝑚2 𝜃 = 𝑡𝑎𝑛−1 ( 𝑚2 − 𝑚1 ) 1 + 𝑚1 𝑚2 Example: Determining the angle between two lines. 𝑦 = −2𝑥 + 2 Sol. From de equations 3 𝑎𝑛𝑑 𝑔 = − 𝑥 − 2 4 3 𝑎𝑛𝑑 𝑔 = − 𝑥 − 2 4 𝑦 = −2𝑥 + 2 𝑚1 = −2 𝑚2 = − 3 4 Applying 3 − 4 − (−2) 𝜃 = 𝑡𝑎𝑛−1 ( ) 3 1 + (− 4)(−2) 3 −4+2 𝜃 = 𝑡𝑎𝑛−1 ( ) 6 1+4 𝜃 = 𝑡𝑎𝑛 −1 −3 + 8 ( 4 ) 10 4 5 𝜃 = 𝑡𝑎𝑛−1 ( 4 ) 5 2 10 𝜃 = 𝑡𝑎𝑛−1 ( ) 20 𝜃 = 𝑡𝑎𝑛−1 (0.5) 𝜃 = 26.57 ° Equation of family of lines A family of lines is a set of lines that have something in common with each other. Straight lines can belong to two types of families: one where the slope (m) is the same one where the “y-intercept” (0, b) is the same The constants of the equation in slope-intercept form "y = mx + b" are “m” and “b”. The values “m” and “b” are fixed for any particular line but change from line to line. These letters are called parameters. Example: Sketch the graph for the lines with the following equations. 𝑦 = −2𝑥 + 2 𝑦 = −𝑥 + 2 𝑦=𝑥+2 𝑦 = 2𝑥 + 2 Example: Sketch the graph for the lines with the following equations. Straight line in polar coordinates Up to this point, we’ve been working with the Cartesian coordinate system. However, this is not the only plane in which we can sketch a point, there is another kind of plane to define a point in a two dimensional space. Instead of moving vertically and horizontally from the origin to get to the point, we could instead go straight out of the origin until hitting the point and then determining the angle that this line makes with the positive x-axis. The coordinates of the point will be the distance of the point from the origin and the rotation from the positive x-axis, that is (𝑟 , 𝜃) Coordinates in this form are called polar coordinates. In polar coordinates the origin is often called the pole. In the previous figure a right triangle is formed, so there are two equations that come from that fact and that will allow converting polar coordinates into Cartesian coordinates and vice versa. Example: Conversions between polar and Cartesian coordinates and vice versa. Convert each of the following points into the given coordinate system, expressing 𝜃 in radians. Sol: Use polar to Cartesian conversion formulas 𝑥 = 𝑟𝑐𝑜𝑠 (𝜃) 2 𝑥 = (−4)𝑐𝑜𝑠 ( 𝜋) 3 𝑥 = (−4)(−0.5) 𝑥=2 𝑦 = 𝑟 𝑠𝑖𝑛 (𝜃) 2 𝑦 = (−4)𝑠𝑖𝑛 ( 𝜋) 3 𝑦 = (−4)(0.8660) 𝑦 = −3.46 So, in Cartesian coordinates this point is ( 2 , −3.46) Sol: Use Cartesian to polar conversion formulas 𝑟 = √𝑥 2 + 𝑦 2 𝑟 = √(−1)2 + (−1)2 𝑟 = √2 𝑦 𝜃 = 𝑡𝑎𝑛−1 ( ) 𝑥 𝜃 = 𝑡𝑎𝑛−1 ( −1 ) −1 𝜃 = 𝑡𝑎𝑛−1 (1) 𝜃 = 0.7853 𝑟𝑎𝑑 Since the point lies in the quadrant III, this value should be added to π 𝜃 = (𝜋 + 0.7853)𝑟𝑎𝑑 𝜃 = (3.1416 + 0.7853)𝑟𝑎𝑑 𝜃 = 3.9270 𝑟𝑎𝑑 So, in polar coordinates the point is ( √2 , 3.9270 )
