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 cos 3x + cos x – cos 2x = 0
Find the general solution for each of the following equations sec2 2x = 1– tan 2x
If \[\tan x = \frac{a}{b},\] show that
If \[cosec x - \sin x = a^3 , \sec x - \cos x = b^3\], then prove that \[a^2 b^2 \left( a^2 + b^2 \right) = 1\]
If \[\sin x + \cos x = m\], then prove that \[\sin^6 x + \cos^6 x = \frac{4 - 3 \left( m^2 - 1 \right)^2}{4}\], where \[m^2 \leq 2\]
Prove that:
Prove that: \[\tan\frac{11\pi}{3} - 2\sin\frac{4\pi}{6} - \frac{3}{4} {cosec}^2 \frac{\pi}{4} + 4 \cos^2 \frac{17\pi}{6} = \frac{3 - 4\sqrt{3}}{2}\]
Prove that:
\[\tan 4\pi - \cos\frac{3\pi}{2} - \sin\frac{5\pi}{6}\cos\frac{2\pi}{3} = \frac{1}{4}\]
If \[cosec x - \cot x = \frac{1}{2}, 0 < x < \frac{\pi}{2},\]
If sec x + tan x = k, cos x =
Which of the following is incorrect?
The value of \[\cos1^\circ \cos2^\circ \cos3^\circ . . . \cos179^\circ\] is
Find the general solution of the following equation:
Find the general solution of the following equation:
Find the general solution of the following equation:
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:
Solve the following equation:
\[5 \cos^2 x + 7 \sin^2 x - 6 = 0\]
Solve the following equation:
cosx + sin x = cos 2x + sin 2x
Write the number of points of intersection of the curves
If \[e^{\sin x} - e^{- \sin x} - 4 = 0\], then x =
Solve the following equations:
2 cos2θ + 3 sin θ – 3 = θ
Solve the following equations:
sin θ + sin 3θ + sin 5θ = 0
Solve the following equations:
sin 2θ – cos 2θ – sin θ + cos θ = θ
Solve the following equations:
cot θ + cosec θ = `sqrt(3)`
Choose the correct alternative:
If sin α + cos α = b, then sin 2α is equal to
Solve 2 tan2x + sec2x = 2 for 0 ≤ x ≤ 2π.
If 2sin2θ = 3cosθ, where 0 ≤ θ ≤ 2π, then find the value of θ.
Find the general solution of the equation sinx – 3sin2x + sin3x = cosx – 3cos2x + cos3x
The minimum value of 3cosx + 4sinx + 8 is ______.
