In math worksheet on rectangular – polar conversion; students can practice the questions on how to convert rectangular coordinates to polar coordinates and also convert polar coordinates to rectangular coordinates (viceversa).
To convert polar coordinates to rectangular coordinates;
x = r cos θ, y = r sin θ
To convert rectangular coordinates to polar coordinates;
r = √(x² + y²) and tan θ = y/x or, θ = tan\(^{1}\) y/x
`To know more about the relation between the Cartesian coordinates and Polar coordinates and about more examples Click Here.
Follow the above formula to solve the below questions given in the worksheet on rectangular – polar conversion.
1. OX and OY are the cartesian axes of coordinates. Again 0 and OX are respectively the pole and initial line of a system of polar coordinates. With respect to these systems (i) if the polar coordinates of a point P be (2, 300), find the cartesian coordinates of the point; (ii) if the cartesian coordinates of a point P be (0, 2), find its polar coordinates.
2. Find the Cartesian coordinates of the points whose polar coordinates are :
(i) (2, π/3)
(ii) (4, 3π/2)
(iii) (6, π/6)
(iv) (4, π/3)
(v) (1, √3).
3. Find the polar coordinates of the points whose Cartesian coordinates are:
(i) (2, 2).
(ii) ( √3, 1)
(iii) ( 1, 1)
(iv) (1,  1)
(v) ((5√3)/2,  5/2).
4. Reduce each of the following Cartesian equations to polar forms:
(i) x² + y² = a²
(ii) y = x tan α
(iii) x cos α + y sin α = p
(iv) y² = 4x + 3
(v) x²  y² = a²
(vi) x² + y² = 2ax
(vii) (x² + y²)² = a²(x²  y²)
5. Transform each of the following polar equations to cartesian forms:
(i) r = 2a sin θ
(ii) l/r = A cos θ + B sin θ
(iii) r= a sin θ
(iv) r² = a²cos 2θ
(v) \(r^{\frac{1}{2}}\) = \(a^{\frac{1}{2}}\) sin θ/2
(vi) r² sin 2θ = 2a²
(vii) r cos (θ  α)
(viii) r(cos 3θ + sin 3θ) = 5k sin θ cos θ.
Answers for the worksheet on rectangular – polar conversion are given below to check the exact answers of the above questions.
1. (i) (√3 ,1)
(ii) (2, π/2);
2. (i) (1, √3)
(ii) (0, 4)
(iii) (3√3, 3)
(iv) (2, 2√3),
(v) (cos √3, sin √3) where √3 is measured in radian.
3.(i) (2√2, π/4)
(ii) (2, 5π/6)
(iii) (√2, 3π/4)
(iv) (√2, π/4)
(v) (5, 7π/6)
4. (i) r² = a²
(ii) θ = α
(iii) r cos (θ  α) = P
(iv) r² sin² θ = 4r cos θ + 3
(v) r² cos 2θ = a²
(vi) r = 2a cos θ
(vii) r² = a² cos 2θ.
5. (i) x² + y² = 2ay
(ii) Ax + By = l
(iii) x² + y² = ay
(iv) (x² + y²)² = a²(x²  y²)
(v) (2x² + 2y² + ax)² = a²(x² + y²)
(vi) xy = a²
(vii) x cos α + y sin α = p
(viii) x³ + 3x²y  3xy²
 y³ = 5kxy.
● Coordinate Geometry
11 and 12 Grade Math
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