Advertisements
Advertisements
प्रश्न
Find the polar coordinates of the point whose Cartesian coordinates are `(1, - sqrt(3))`.
Advertisements
उत्तर
Here x = 1 and y = `- sqrt(3)`
∴ The point lies in the fourth quadrant.
Let the polar coordinates be (r, θ).
Then r2 = x2 + y2 = `(1)^2 + (- sqrt(3))^2` = 1 + 3 = 4
∴ r = 2 ...[ ∵ r > 0]
`cos θ = x/r = (1)/(2)`
and `sin θ = y/r = - sqrt(3)/(2)`
∴ tan θ = `- sqrt(3)`
Since, the point lies in the fourth quadrant and 0 ≤ θ < 2π
tan θ = `- sqrt(3) = - tan π/3`
= `tan(2π - π/3)` ...[ ∵ tan(2π – θ) = –tan θ]
= `tan (5π)/3`
∴ θ = `(5π)/3`
∴ The polar coordinates of the given point are `(2, (5π)/3)`.
संबंधित प्रश्न
In Δ ABC, if a = 13, b = 14 and c = 15, then sin (A/2)= _______.
(A) `1/5`
(B) `sqrt(1/5)`
(C) `4/5`
(D) `2/5`
The angles of the ΔABC are in A.P. and b:c=`sqrt3:sqrt2` then find`angleA,angleB,angleC`
If in ∆ABC with usual notations a = 18, b = 24, c = 30 then sin A/2 is equal to
(A) `1/sqrt5`
(B) `1/sqrt10`
(C) `1/sqrt15`
(D) `1/(2sqrt5)`
The principal solutions of cot x = -`sqrt3` are .................
In , ΔABC prove that
`"sin"(("B" - "C")/2) = (("b" - "c")/"a") "cos"("A"/2)`
In ,Δ ABC with usual notations prove that
b2 = c2 +a2 - 2 ca cos B
Find the Cartesian co-ordinates of the point whose polar co-ordinates are:
`(sqrt(2), pi/4)`
Find the polar co-ordinates of the point whose Cartesian co-ordinates are.
`(0, 1/2)`
Solve the triangle in which a = `(sqrt3 + 1)`, b = `(sqrt3 - 1)` and ∠C = 60°.
In Δ ABC, if ∠C = 90°, then prove that sin (A - B) = `("a"^2 - "b"^2)/("a"^2 + "b"^2)`
In ΔABC, if `"cos A"/"a" = "cos B"/"b"`, then show that it is an isosceles triangle.
In Δ ABC, if sin2 A + sin2 B = sin2 C, then show that the triangle is a right-angled triangle.
In Δ ABC, if a cos2 `"C"/2 + "c cos"^2 "A"/2 = "3b"/2`, then prove that a, b, c are in A.P.
Show that `(9pi)/8 - 9/4 sin^-1 (1/3) = 9/4 sin^-1 ((2sqrt2)/3)`.
If sin `(sin^-1 1/5 + cos^-1 x) = 1`, then find the value of x.
If `tan^-1 (("x" - 1)/("x" - 2)) + tan^-1 (("x" + 1)/("x" + 2)) = pi/4`, find the value of x.
State whether the following equation has a solution or not?
cos 2θ = `1/3`
In ∆ABC, if cos A = `(sinB)/(2sinC)`, then ∆ABC is ______.
In ∆ABC, if b2 + c2 − a2 = bc, then ∠A = ______.
Find the Cartesian co-ordinates of point whose polar co-ordinates are `(4, pi/3)`
In ΔABC, if a cos A = b cos B, then prove that ΔABC is either a right angled or an isosceles triangle.
In ∆ABC, prove that `sin ((A - B)/2) = ((a - b)/c) cos C/2`
If the angles A, B, C of ΔABC are in A.P. and its sides a, b, c are in G.P., then show that a2, b2, c2 are in A.P.
In ∆ABC, prove that `(cos^2"A" - cos^2"B")/("a" + "b") + (cos^2"B" - cos^2"C")/("b" + "c") + (cos^2"C" - cos^2"A")/("c" + "a")` = 0
In ∆ABC, if ∠A = `pi/2`, then prove that sin(B − C) = `("b"^2 - "c"^2)/("b"^2 + "c"^2)`
In ΔABC, if (a+ b - c)(a + b + c) = 3ab, then ______.
In a triangle ABC with usual notations, if `(cos "A")/"a" = (cos "B")/"b" = (cos "C")/"c"`, then area of triangle ABC with a = `sqrt6` is ____________.
In a ΔABC, `(sin "C"/2)/(cos(("A" - "B")/2))` = ______
In a ΔABC, 2ab sin`((A + B - C)/2)` = ______
If one side of a triangle is double the other and the angles opposite to these sides differ by 60°, then the triangle is ______
If `(- sqrt2, sqrt2)` are cartesian co-ordinates of the point, then its polar co-ordinates are ______.
If P(6, 10, 10), Q(1, 0, -5), R(6, -10, λ) are vertices of a triangle right angled at Q, then value of λ is ______.
The polar co-ordinates of P are `(2, pi/6)`. If Q is the image of P about the X-axis then the polar co-ordinates of Q are ______.
In ΔABC if sin2A + sin2B = sin2C and l(AB) = 10, then the maximum value of the area of ΔABC is ______
If polar co-ordinates of a point are `(1/2, pi/2)`, then its cartesian co-ordinates are ______.
In a ΔABC, if a = `sqrt(2)` x and b = 2y and ∠C = 135°, then the area of triangle is ______.
In ΔABC with usual notations, if ∠A = 30° and a = 5, then `s/(sumsinA)` is equal to ______.
The number of solutions of the equation sin 2x – 2 cosx + 4 sinx = 4 in the interval [0, 5π] is ______.
If in a ΔABC `a cos^2(C/2) + c cos^2(A/2) = (3b)/2`, then the sides a, b and c ______.
In ΔABC, with usual notations, if a, b, c are in A.P. Then `a cos^2 (C/2) + c cos^2(A/2)` = ______.
If in ΔABC, `sin A/2 * sin C/2 = sin B/2` and 2s is the perimeter of the triangle, then s = ______.
