Name: Introduction to Half-angle Formulae
Uploaded: 2019-07-19
Duration: 4 min 47 s
Description: This video introduces the half-angle formulae, which will help you solve challenging questions in the exams. Learn more through this video.

Question

If in an obtuse angled triangle the obtuse angle is

\frac{3 π}{4}

and the other two angles are equal to two values of

θ

satisfying

a \tan θ + b \sec θ = c,

where

| b | \leq \sqrt{a^{2} + c^{2}},

then find the value of

a^{2} - c^{2}

.

Accepted Answer

In this question we see that we have $tanx, secx$ which are related as follows $\sec^{2} x = 1 + \tan^{2} x$ and we know that we do not require $b$ :

$a \tan θ + b \sec θ = c$

$\Rightarrow b \sec θ = c - a \tan θ$

On squaring both sides:

$\Rightarrow b^{2} \sec^{2} θ = c^{2} - 2 a c \tan θ + a^{2} \tan^{2} θ$

On dividing both sides by $b^{2}$ and taking $\sec^{2} θ = 1 + \tan^{2} θ$

$\Rightarrow 1 + \tan^{2} θ = \frac{c^{2}}{b^{2}} - 2 \frac{a c}{b^{2}} \tan θ + \frac{a^{2}}{b^{2}} \tan^{2} θ$

$| b | \leq \sqrt{a^{2} + c^{2}},$

Taking $\frac{1}{b^{2}}$ common and equating it, we get:

$\Rightarrow (a^{2} - b^{2}) \tan^{2} θ - 2 a c \tan θ + c^{2} - b^{2} = 0 \dots (1)$

Let us assume the two angles satisfying the above equation are: $A, B$

It is given in the question that the obtuse angle is $\frac{3 π}{4}$ , $A + B = \frac{π}{4}$

We have sum of roots: $- \frac{b}{a}$ and product of roots: $\frac{c}{a}$ in a quadratic equation $a x^{2} + b x + c = 0$

Now from $(1)$ we know that:

$\tan A + \tan B = \frac{2 a c}{a^{2} - b^{2}}$ and $\tan A \cdot \tan B = \frac{c^{2} - b^{2}}{a^{2} - b^{2}}$

and $\tan (A + B) = \tan \frac{π}{4} = 1$ , $\tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A . \tan B}$

$\Rightarrow 1 - \tan A . \tan B = \tan A + \tan B$

$\Rightarrow 1 - \frac{c^{2} - b^{2}}{a^{2} - b^{2}} = \frac{2 a c}{a^{2} - b^{2}}$

$\Rightarrow \frac{a^{2} - b^{2} - c^{2} + b^{2}}{a^{2} - b^{2}} = \frac{2 a c}{a^{2} - b^{2}}$ $\Rightarrow a^{2} - c^{2} = 2 a c$

If in an obtuse angled triangle the obtuse angle is $\frac{3 π}{4}$ and the other two angles are equal to two values of $θ$ satisfying $a \tan θ + b \sec θ = c,$ where $| b | \leq \sqrt{a^{2} + c^{2}},$ then find the value of $a^{2} - c^{2}$ .

Important Questions on Properties and Solutions of Triangles

Learn and Prepare for any exam you want !

If in an obtuse angled triangle the obtuse angle is 3π4 and the other two angles are equal to two values of θ satisfying atanθ+bsecθ=c, where |b|≤a2+c2, then find the value of a2-c2.

Important Questions on Properties and Solutions of Triangles

Learn and Prepare for any exam you want !

If in an obtuse angled triangle the obtuse angle is $\frac{3 π}{4}$ and the other two angles are equal to two values of $θ$ satisfying $a \tan θ + b \sec θ = c,$ where $| b | \leq \sqrt{a^{2} + c^{2}},$ then find the value of $a^{2} - c^{2}$ .