If a cos 2 - b sin 2 - c has - and - as its roots- that prove thati tan-tan-2 b - a - cii tan-tan- c - a - c - aiii tan -b-a

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Equations of the Form acosθ+bsinθ = c

If a cos 2 θ+...

Question

If $a c o s 2 θ + b s i n 2 θ = c$ has $α a n d β$ as its roots, that prove that

(i) $t a n α + t a n β = \frac{2 b}{a + c}$

(ii) $t a n α t a n β = \frac{c - a}{c + a}$

(iii) $t a n (α + β) = \frac{b}{a}$

Open in App

Solution

$c o s 2 θ = \frac{1 - t a n^{2} θ}{1 + t a n^{2} θ} s i n 2 θ = \frac{2 t a n θ}{1 + t a n^{2} θ}$

substitute these values in the given equation, it reduces to $a (1 - t a n^{2} θ) + b (2 t a n θ) = c (1 + t a n^{2} θ)$

$(c + a) t a n^{2} θ + 2 b t a n θ + c - a = 0$

As $α a n d β$ are roots

sum of the roots, $t a n α + t a n β = \frac{2 b}{c + a}$

Product of roots, $t a n α t a n β = \frac{c - a}{c + a}$

$t a n (α + β) = \frac{t a n α + t a n β}{1 - t a n α t a n β} = \frac{2 b}{c + a - c + a} = \frac{b}{a}$

Suggest Corrections

Similar questions

Q. If

a \cos 2 x + b \sin 2 x = c

has α and β as its roots, then prove that

(i)

\tan α + \tan β = \frac{2 b}{a + c}

[NCERT EXEMPLAR]

(ii)

\tan α \tan β = \frac{c - a}{c + a}

(iii)

\tan (α + β) = \frac{b}{a}

[NCERT EXEMPLAR]

Q. If

a cos 2 θ + b sin 2 θ = c

has

α

and

β

as its solutions ,then show that

tan α + tan β = \frac{2 b}{c + a}, tan α tan β = \frac{c - a}{c + a}

If $a c o s 2 θ + b s i n 2 θ = c$ has $α$ and $β$ as its roots, prove that $α + t a n β = \frac{2 b}{a + c}$ and $α - t a n β = \frac{2}{a + c} \sqrt{b^{2} - c^{2} + a^{2} .}$

Q. If

a cos 2 θ + b sin 2 θ = c

has

α

and

β

as its roots then

tan α + tan β = \frac{2 b}{a + c}

State whether true or false.

Q. If

a c o s 2 θ + b s i n 2 θ = c h a s α a n d β

as its solution, then the value of

t a n α + t a n β

Join BYJU'S Learning Program

If a cos 2θ+b sin 2θ=c has α and β as its roots, that prove that (i) tan α+tan β=2ba+c (ii) tan α tan β=c−ac+a (iii) tan (α+β)=ba

If $a c o s 2 θ + b s i n 2 θ = c$ has $α a n d β$ as its roots, that prove that

(i) $t a n α + t a n β = \frac{2 b}{a + c}$

(ii) $t a n α t a n β = \frac{c - a}{c + a}$

(iii) $t a n (α + β) = \frac{b}{a}$