Advertisements
Advertisements
प्रश्न
In a triangle ABC with ∠C = 90° the equation whose roots are tan A and tan B is ______.
Advertisements
उत्तर
In a triangle ABC with ∠C = 90° the equation whose roots are tan A and tan B is `underline(x^2 - (2/(sin 2A)) x + 1` = 0.
Explanation:
Given a ΔABC with ∠C = 90°
So, the equation whose roots are tanA and tanB is
x2 – (tanA + tanB)x + tanA.tanB = 0
A + B = 90° ......[∵ ∠C = 90°]
⇒ tan(A + B) = tan90°
⇒ `(tanA + tanB)/(1 - tanA tanB) = 1/0`
⇒ 1 – tanA tanB = 0
⇒ tan A tan B = 1 .......(i)
Now tanA + tanB = `sinA/cosA + sinB/cosB`
= `(sinA cosB + cosA sinB)/(cosA cosB)`
= `(sin(A + B))/(cosA cosB)`
= `(sin 90^circ)/(cosA. cos(90^circ - A))`
= `1/(cosA sinA)`
∴ tanA + tanB = `2/(2sinA cosA)`
= `2/(sin 2A)` ......(ii)
From (i) and (ii) we get
`x^2 - (2/(sin 2A)) x + 1` = 0
APPEARS IN
संबंधित प्रश्न
Find the general solution of the equation sin 2x + cos x = 0
If \[T_n = \sin^n x + \cos^n x\], prove that \[\frac{T_3 - T_5}{T_1} = \frac{T_5 - T_7}{T_3}\]
If \[T_n = \sin^n x + \cos^n x\], prove that \[6 T_{10} - 15 T_8 + 10 T_6 - 1 = 0\]
Prove that: tan 225° cot 405° + tan 765° cot 675° = 0
Prove that:
\[\frac{\cos (2\pi + x) cosec (2\pi + x) \tan (\pi/2 + x)}{\sec(\pi/2 + x)\cos x \cot(\pi + x)} = 1\]
Prove that
Prove that
If tan x = \[x - \frac{1}{4x}\], then sec x − tan x is equal to
sin6 A + cos6 A + 3 sin2 A cos2 A =
If \[cosec x + \cot x = \frac{11}{2}\], then tan x =
The value of sin25° + sin210° + sin215° + ... + sin285° + sin290° is
sin2 π/18 + sin2 π/9 + sin2 7π/18 + sin2 4π/9 =
If x sin 45° cos2 60° = \[\frac{\tan^2 60^\circ cosec30^\circ}{\sec45^\circ \cot^{2^\circ} 30^\circ}\], then x =
If A lies in second quadrant 3tan A + 4 = 0, then the value of 2cot A − 5cosA + sin A is equal to
The value of \[\cos1^\circ \cos2^\circ \cos3^\circ . . . \cos179^\circ\] is
The value of \[\tan1^\circ \tan2^\circ \tan3^\circ . . . \tan89^\circ\] is
Find the general solution of the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
`cosec x = 1 + cot x`
Solve the following equation:
\[5 \cos^2 x + 7 \sin^2 x - 6 = 0\]
Solve the following equation:
\[\sin x - 3\sin2x + \sin3x = \cos x - 3\cos2x + \cos3x\]
Solve the following equation:
\[2^{\sin^2 x} + 2^{\cos^2 x} = 2\sqrt{2}\]
Write the number of solutions of the equation
\[4 \sin x - 3 \cos x = 7\]
Write the values of x in [0, π] for which \[\sin 2x, \frac{1}{2}\]
and cos 2x are in A.P.
If \[\tan px - \tan qx = 0\], then the values of θ form a series in
If a is any real number, the number of roots of \[\cot x - \tan x = a\] in the first quadrant is (are).
Find the principal solution and general solution of the following:
cot θ = `sqrt(3)`
Solve the following equations:
`tan theta + tan (theta + pi/3) + tan (theta + (2pi)/3) = sqrt(3)`
Choose the correct alternative:
If sin α + cos α = b, then sin 2α is equal to
Solve the equation sin θ + sin 3θ + sin 5θ = 0
If a cosθ + b sinθ = m and a sinθ - b cosθ = n, then show that a2 + b2 = m2 + n2
Find the general solution of the equation sinx – 3sin2x + sin3x = cosx – 3cos2x + cos3x
The minimum value of 3cosx + 4sinx + 8 is ______.
