In a - ABC- if cos A -sin A -2 sin C- show that the triangle is isosceles-

By the sine rule, we have a/sin A = b/sin B = c/sin C ⇒sin A/a = sin B/b = sin C/c = k (say) ⇒ sin A = ka, sin B = kb and sin C = kc. ∴ cos A = sin B/2 sin C⇒ ...

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

Byju's Answer

Standard XI

Mathematics

Trigonometric Ratios of Multiples of an Angle

In a Δ ABC, i...

Question

In a $Δ A B C,$ if cos A = $\frac{s i n A}{2 s i n C},$ show that the triangle is isosceles.

Open in App

Solution

By the sine rule, we have

$\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}$

$\Rightarrow \frac{s i n A}{a} = \frac{s i n B}{b} = \frac{s i n C}{c} = k (s a y)$

$\Rightarrow s i n A = k a, s i n B = k b a n d s i n C = k c .$

$∴ c o s A = \frac{s i n B}{2 s i n C} \Rightarrow \frac{(b^{2} + c^{2} - a^{2})}{2 b c} = \frac{k b}{2 k c}$

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

$\Rightarrow c^{2} = a^{2} \Rightarrow | c | = | a |$

$\Rightarrow A B = B C .$

Hence, $Δ A B C .$ is isosceles.

Suggest Corrections

Similar questions

$I n a Δ A B C, i f c o s C = \frac{s i n A}{2 s i n B}, prove that the triangle is isosceles.$

Q. If in a

△ A B C, cos A = \frac{sin B}{2 sin C}

, prove that it is an isosceles triangle.

Q. In

Δ A B C,

\frac{cos A}{a} = \frac{cos B}{b},

then show that it is an isosceles triangle.

In a $Δ A B C,$ if a cos A = b cos B, show that the triangle is either isosceles or right - angled.

Q. If

cos A = sin B / (2 sin C)

, prove that

△ A B C

is isosceles.

Join BYJU'S Learning Program

In a ΔABC, if cos A = sinA2sinC, show that the triangle is isosceles.

By the sine rule, we have asinA=bsinB=csinC ⇒sinAa=sinBb=sinCc=k(say) ⇒sin A=ka,sin B=kb and sin C=kc. ∴cosA=sinB2sinC⇒(b2+c2−a2)2bc=kb2kc ⇒(b2+c2−a2)=b2 ⇒c2=a2⇒|c|=|a| ⇒AB=BC. Hence, ΔABC. is isosceles.

In a $Δ A B C,$ if cos A = $\frac{s i n A}{2 s i n C},$ show that the triangle is isosceles.