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Trigonometry: Polar Coordinates Honors Precalculus Mr. Velazquez Plotting Points with Polar Coordinates Plotting Points with Polar Coordinates y Plot the points with the following polar coordinates: (a) (4, 30°) (b) 3𝜋 3, 4 (c) (−2, 120°) x Representations of Points in Polar Coords Representations of Points in Polar Coords 𝜋 The point 3, 6 is plotted in the figure. Find another representation of the point in which: (a) r is positive, and 2𝜋 ≤ 𝜃 < 4𝜋 y (b) r is negative, and 0 ≤ 𝜃 < 2𝜋 x (c) r is negative, and −2𝜋 ≤ 𝜃 < 0 Polar and Rectangular Coordinates Polar and Rectangular Coordinates Polar and Rectangular Coordinates Find the rectangular coordinates of the points with the following polar coordinates: (a) y 𝜋 3, 2 x (b) 2𝜋 −2, 3 Polar and Rectangular Coordinates Convert the rectangular point (−1, 3) to polar coordinates: Polar and Rectangular Coordinates Find the polar coordinates of the points with the following rectangular coordinates: (a) y 3 , −1 x (b) −2, 2 3 Equation Conversion: Rectangular to Polar We can turn an equation in rectangular form to an equation in polar form in the following steps: 1. Substitute 𝒙 = 𝒓 𝐜𝐨𝐬 𝜽 and 𝒚 = 𝒓 𝐬𝐢𝐧 𝜽 for 𝑥 and 𝑦. 2. Solve the resulting equation for 𝑟. Convert the rectangular equation to a polar equation that expresses r in terms of . y 3x 2 Equation Conversion: Rectangular to Polar We can turn an equation in rectangular form to an equation in polar form in the following steps: 1. Substitute 𝒙 = 𝒓 𝐜𝐨𝐬 𝜽 and 𝒚 = 𝒓 𝐬𝐢𝐧 𝜽 for 𝑥 and 𝑦. 2. Solve the resulting equation for 𝑟. Convert the rectangular equation to a polar equation that expresses r in terms of . ( x 3) 2 +(y+2) 2 1 Equation Conversion: Polar to Rectangular We can turn an equation in polar form to an equation in rectangular form with the following substiutions: 𝑟2 = 𝑥2 + 𝑦2 𝑟 cos 𝜃 = 𝑥 𝑟 sin 𝜃 = 𝑦 Convert the polar equation to a rectangular equation in x and y. r=3 tan 𝜃 = 𝑦 𝑥 Equation Conversion: Polar to Rectangular We can turn an equation in polar form to an equation in rectangular form with the following substiutions: 𝑟2 = 𝑥2 + 𝑦2 𝑟 cos 𝜃 = 𝑥 𝑟 sin 𝜃 = 𝑦 Convert the polar equation to a rectangular equation in x and y. = 6 tan 𝜃 = 𝑦 𝑥 Equation Conversion: Polar to Rectangular We can turn an equation in polar form to an equation in rectangular form with the following substiutions: 𝑟2 = 𝑥2 + 𝑦2 𝑟 cos 𝜃 = 𝑥 𝑟 sin 𝜃 = 𝑦 Convert the polar equation to a rectangular equation in x and y. r= -2 cosx tan 𝜃 = 𝑦 𝑥 Equation Conversion: Polar to Rectangular We can turn an equation in polar form to an equation in rectangular form with the following substiutions: 𝑟2 = 𝑥2 + 𝑦2 𝑟 cos 𝜃 = 𝑥 𝑟 sin 𝜃 = 𝑦 Convert the polar equation to a rectangular equation in x and y. r= 5 csc tan 𝜃 = 𝑦 𝑥 Exit Ticket: Polar Coordinates 1. Convert the following equations from rectangular to polar form: 2. Convert the following equations from polar to rectangular form: HOMEWORK (DUE 3/10) Pg. 672-673, #4-72 (multiples of 4)