Advertisements
Advertisements
प्रश्न
In Δ ABC, if sin2 A + sin2 B = sin2 C, then show that the triangle is a right-angled triangle.
Advertisements
उत्तर
By sine rule,
`"sin A"/"a" = "sin B"/"b" = "sin C"/"c "` = k
∴ sin A = ka, sin B = kb, sin C = kc
∴ sin2A + sin2B = sin2C
∴ k2a2 + k2b2 = k2c2
∴ a2 + b2 = c2
∴ Δ ABC is a rightangled triangle, rightangled at C.
APPEARS IN
संबंधित प्रश्न
In any ΔABC if a2 , b2 , c2 are in arithmetic progression, then prove that Cot A, Cot B, Cot C are in arithmetic progression.
In ΔABC, prove that `tan((A - B)/2) = (a - b)/(a + b)*cot C/2`.
In any ΔABC, with usual notations, prove that b2 = c2 + a2 – 2ca cos B.
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)`
With usual notations, in ΔABC, prove that a(b cos C − c cos B) = b2 − c2
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 Cartesian coordinates of the point whose polar coordinates are :
`(4, pi/2)`
Find the polar co-ordinates of the point whose Cartesian co-ordinates are.
`(sqrt(2), sqrt(2))`
Find the polar co-ordinates of the point whose Cartesian co-ordinates are.
`(3/2, (3√3)/2)`.
In ΔABC, if cot A, cot B, cot C are in A.P. then show that a2, b2, c2 are also in A.P.
In any ΔABC, prove the following:
`("c" - "b cos A")/("b" - "c cos A") = ("cos B")/("cos C")`
In any Δ ABC, prove the following:
ac cos B - bc cos A = a2 - b2
In any Δ ABC, prove the following:
`("b" - "c")/"a" = (tan "B"/2 - tan "C"/2)/(tan "B"/2 +tan "C"/2)`
In Δ ABC, if ∠C = 90°, then prove that sin (A - B) = `("a"^2 - "b"^2)/("a"^2 + "b"^2)`
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.
Prove that `tan^-1 sqrt"x" = 1/2 cos^-1 ((1 - "x")/(1 + "x"))`, if x ∈ [0, 1]
State whether the following equation has a solution or not?
cos 2θ = `1/3`
In ∆ABC, if b2 + c2 − a2 = bc, then ∠A = ______.
If polar co-ordinates of a point are `(3/4, (3pi)/4)`, then its Cartesian co-ordinate are ______
With usual notations, prove that `(cos "A")/"a" + (cos "B")/"b" + (cos "C")/"c" = ("a"^2 + "b"^2 + "c"^2)/(2"abc")`
In ∆ABC, prove that `("b" - "c")^2 cos^2 ("A"/2) + ("b" + "c")^2 sin^2 ("A"/2)` = a2
In ∆ABC, if a = 13, b = 14, c = 15, then find the value of cos B
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 ΔABC, prove that `("b"^2 - "c"^2)/"a" cos"A" + ("c"^2 - "a"^2)/"b" cos"B" + ("a"^2 - "b"^2)/"c" cos "C"` = 0
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 Δ ABC; with usual notations, if cos A = `(sin "B")/(sin "C")`, then the triangle is _______.
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 ______
In ΔABC if sin2A + sin2B = sin2C and l(AB) = 10, then the maximum value of the area of ΔABC is ______
The smallest angle of the ΔABC, when a = 7, b = `4sqrt(3)` and c = `sqrt(13)` is ______.
If polar co-ordinates of a point are `(1/2, pi/2)`, then its cartesian co-ordinates are ______.
If in a `triangle"ABC",` a2cos2 A - b2 - c2 = 0, then ______.
In ΔABC, if `"a" cos^2 "C"/2 + "c" cos^2 "A"/2 = (3"b")/2`, then a, b, c are in ______.
In a triangle ABC, b = `sqrt3`, c = 1 and ∠A = 30°, then the largest angle of the triangle is ______
If in a triangle ABC, AB = 5 units, AB = 5 units, ∠B = `cos^-1 (3/5)` and radius of circumcircle of ΔABC is 5 units, then the area (in sq.units) of ΔABC 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 a triangle ABC, ∠C = 90°, then `(a^2 - b^2)/(a^2 + b^2)` is ______.
In ΔABC, a = 3, b = 1, cos(A – B) = `2/9`, find c.
