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 (vice-versa).
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 co-ordinates and Polar co-ordinates 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 co-ordinates. Again 0 and OX are respectively the pole and initial line of a system of polar co-ordinates. With respect to these systems (i) if the polar co-ordinates of a point P be (2, 300), find the cartesian co-ordinates of the point; (ii) if the cartesian co-ordinates of a point P be (0, 2), find its polar co-ordinates.
2. Find the Cartesian co-ordinates of the points whose polar co-ordinates are :
(i) (2, π/3)
(ii) (4, 3π/2)
(iii) (6, -π/6)
(iv) (-4, π/3)
(v) (1, √3).
3. Find the polar co-ordinates of the points whose Cartesian co-ordinates 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.
● Co-ordinate Geometry
11 and 12 Grade Math
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