Advertisements
Advertisements
प्रश्न
If `(sin "A")/(sin "C") = (sin ("A - B"))/(sin ("B - C"))`, then show that a2, b2, c2 are in A.P.
Advertisements
उत्तर
By sine rule,
`"sin A"/"a" = "sin B"/"b" = "sin C"/"c" = "k"`
∴ sin A = ka, sin B = kb, sin C = kc
Now, `(sin "A")/(sin "C") = (sin ("A - B"))/(sin ("B - C"))`
∴ sin A . sin (B - C) = sin C. sin (A - B)
∴ sin [π - (B + C)]. sin (B - C)
= sin [π - (A + B)]. sin(A - B) .....[∵ A + B + C = π]
∴ sin (B + C). sin (B - C) = sin (A + B). sin (A - B)
∴ sin2B - sin2C = sin2A - sin2B
∴ 2 sin2B = sin2A + sin2C
∴ 2k2b2 = k2a2 + k2c2
∴ 2b2 = a2 + c2
Hence, a2, b2, c2 are in A.P.
APPEARS IN
संबंधित प्रश्न
Find the principal solution of the following equation:
cosθ = `(1)/(2)`
Find the principal solution of the following equation :
cot θ = `sqrt(3)`
Find the principal solution of the following equation:
sin θ = `-1/2`
Find the general solution of the following equation:
tan θ = `(1)/(sqrt(3))`
Find the general solution of the following equation:
cot 4θ = – 1
Find the general solution of the following equation:
4 cos2 θ = 3
State whether the following equation have solution or not?
3 tanθ = 5
In ΔABC, if ∠A = 45°, ∠B = 60° then find the ratio of its sides.
With the usual notations prove that `2{asin^2 "C"/(2) + "c"sin^2 "A"/(2)}` = a – b + c.
With usual notations prove that 2(bc cosA + ac cosB + ab cosC) = a2 + b2 + c2 .
Select the correct option from the given alternatives:
In Δ ABC if ∠A = 45°, ∠B = 30°, then the ratio of its sides are
If `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha`, then α = ______.
If tan θ + tan 2θ + tan 3θ = tan θ.tan 2θ. tan 3θ, then the general value of the θ is ______.
Find the general solutions of the following equation:
`tan^2 theta = 3`
In Δ ABC, prove that `cos(("A" - "B")/2) = (("a" + "b")/"c")sin "C"/2` .
In ΔABC, prove that `("a - b")^2 cos^2 "C"/2 + ("a + b")^2 sin^2 "C"/2 = "c"^2`
If 2 tan-1(cos x) = tan-1(2 cosec x), then find the value of x.
Prove the following:
`cos^-1 "x" = tan^-1 (sqrt(1 - "x"^2)/"x")`, if x > 0
Prove the following:
`cos^-1 "x" = pi + tan^-1 (sqrt(1 - "x"^2)/"x")`, if x < 0
The principal solutions of `sqrt(3)` sec x − 2 = 0 are ______
Find the principal solutions of cosec x = 2
Find the principal solutions of cos 2𝑥 = 1
If cos–1x + cos–1y – cos–1z = 0, then show that x2 + y2 + z2 – 2xyz = 1
If `|bar"a"|` = 10, `|bar"b"| = 2`, then `sqrt(|bar"a" xx bar"b"|^2 + |bar"a"*bar"b"|^2)` = ?
The general solution of sec θ = `sqrt2` is
`int (sin (log x))^2/x` log x dx = ?
If tan-1 x + 2cot-1 x = `(5pi)/6`, then x is
`[sin (tan^-1 3/4)]^2 + [sin(tan^-1 4/3)]^2 = ?`
If x + y = `pi/2`, then the maximum value of sin x. sin y is.
If y = sin-1 `[(sqrt(1 + x) + sqrt(1 - x))/2]`, then `"dy"/"dx"` = ?
If 2 cos2 θ + 3 cos θ = 2, then permissible value of cos θ is ________.
The values of x in `(0, pi/2)` satisfying the equation sin x cos x = `1/4` are ______.
The value of θ in (π, 2π) satisfying the equation sin2θ - cos2θ = 1 is ______
Which of the following is true in a triangle ABC?
The general solution of cot θ + tan θ = 2 is ______.
If `(tan 3 theta - 1)/(tan 3 theta + 1) = sqrt3`, then the general value of θ is ______.
The general solution of 4sin2 x = 1 is ______.
Find the principal solutions of cot θ = 0
The general solution of the equation tan θ + tan 4θ + tan 7θ = tan θ tan 4θ tan 7θ is ______.
The general solution of sin x – 3 sin 2x + sin 3x = cos x – 3 cos 2x + cos 3x is ______.
With usual notations, in any ΔABC, if a cos B = b cos A, then the triangle is ______.
Find the general solution of sin θ + sin 3θ + sin 5θ = 0
The general solution of cot 4x = –1 is ______.
If tan θ + sec θ = `sqrt(3)`, find the general value of θ.
