Advertisements
Advertisements
Question
In a triangle ABC with ∠C = 90° the equation whose roots are tan A and tan B is ______.
Advertisements
Solution
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
RELATED QUESTIONS
Find the principal and general solutions of the equation `tan x = sqrt3`
Find the general solution of cosec x = –2
Find the general solution of the equation cos 4 x = cos 2 x
Find the general solution of the equation sin 2x + cos x = 0
If \[\tan x = \frac{a}{b},\] show that
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:
Prove that:
\[\frac{\cos (2\pi + x) cosec (2\pi + x) \tan (\pi/2 + x)}{\sec(\pi/2 + x)\cos x \cot(\pi + x)} = 1\]
Find x from the following equations:
\[x \cot\left( \frac{\pi}{2} + \theta \right) + \tan\left( \frac{\pi}{2} + \theta \right)\sin \theta + cosec\left( \frac{\pi}{2} + \theta \right) = 0\]
Prove that:
\[\tan 4\pi - \cos\frac{3\pi}{2} - \sin\frac{5\pi}{6}\cos\frac{2\pi}{3} = \frac{1}{4}\]
If tan x + sec x = \[\sqrt{3}\], 0 < x < π, then x is equal to
If tan \[x = - \frac{1}{\sqrt{5}}\] and θ lies in the IV quadrant, then the value of cos x is
If \[cosec x - \cot x = \frac{1}{2}, 0 < x < \frac{\pi}{2},\]
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
If \[cosec x + \cot x = \frac{11}{2}\], then tan x =
If tan θ + sec θ =ex, then cos θ equals
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:
\[\sin^2 x - \cos x = \frac{1}{4}\]
Solve the following equation:
Solve the following equation:
\[\sqrt{3} \cos x + \sin x = 1\]
Solve the following equation:
`cosec x = 1 + cot x`
Solve the following equation:
\[\sec x\cos5x + 1 = 0, 0 < x < \frac{\pi}{2}\]
Solve the following equation:
3sin2x – 5 sin x cos x + 8 cos2 x = 2
Solve the following equation:
\[2^{\sin^2 x} + 2^{\cos^2 x} = 2\sqrt{2}\]
The smallest value of x satisfying the equation
If \[\cos x + \sqrt{3} \sin x = 2,\text{ then }x =\]
If \[\tan px - \tan qx = 0\], then the values of θ form a series in
If \[\cos x = - \frac{1}{2}\] and 0 < x < 2\pi, then the solutions are
Find the principal solution and general solution of the following:
sin θ = `-1/sqrt(2)`
Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°
2 cos2x + 1 = – 3 cos x
Solve the following equations:
`sin theta + sqrt(3) cos theta` = 1
Solve the equation sin θ + sin 3θ + sin 5θ = 0
