If in a Delta ABC,cos A cos B + sin A sin B s | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given $\cos A \cos B + \sin A \sin B \sin^2 C = 1$.Since $\sin^2 C \le 1$,we have $\cos A \cos B + \sin A \sin B \sin^2 C \le \cos A \cos B + \sin

Vedclass · Accepted Answer

$\Delta ABC$ is isosceles but not right-angled

Vedclass · Answer

(A) Given $\cos A \cos B + \sin A \sin B \sin^2 C = 1$.Since $\sin^2 C \le 1$,we have $\cos A \cos B + \sin A \sin B \sin^2 C \le \cos A \cos B + \sin A \sin B = \cos(A - B)$.Thus,$\cos(A - B) \ge 1$,which implies $\cos(A - B) = 1$,so $A = B$.Substituting $A = B$ into the original equation: $\cos^2 A + \sin^2 A \sin^2 C = 1$.$\sin^2 A \sin^2 C = 1 - \cos^2 A = \sin^2 A$.Since $A$ is an angle of a triangle,$\sin A 
eq 0$,so $\sin^2 C = 1$,which means $C = \frac{\pi}{2}$.Since $A + B + C = \pi$ and $A = B$,we have $2A + \frac{\pi}{2} = \pi$,so $A = B = \frac{\pi}{4}$.Therefore,the triangle is an isosceles right-angled triangle with angles $(\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2})$.Statement $(A)$ is incorrect because the triangle is right-angled.

If in a $\Delta ABC$,$\cos A \cos B + \sin A \sin B \sin^2 C = 1$,then the statement which is incorrect is:

Similar Questions

The sides of a triangle inscribed in a given circle subtend angles $\alpha, \beta, \gamma$ at the center. The minimum value of the $A.M.$ of $\cos (\alpha + \frac{\pi}{2})$,$\cos (\beta + \frac{\pi}{2})$ and $\cos (\gamma + \frac{\pi}{2})$ is equal to

In $\Delta ABC$,if $\sin A : \sin C = \sin (A - B) : \sin (B - C)$,then

In a $\triangle ABC$,the expression $\frac{a}{\tan A} + \frac{b}{\tan B} + \frac{c}{\tan C}$ is equal to:

If angles $A$,$B$,and $C$ of a $\Delta ABC$ are $75^o$,$45^o$,and $60^o$ respectively,then the ratio of the areas of $\Delta OBC$,$\Delta COA$,and $\Delta AOB$ respectively is [where $O$ is the circumcentre of the triangle].

In $\Delta ABC$,if $\cos A + \cos C = 4 \sin^2 \frac{B}{2}$,then $a, b, c$ are in

Vedclass Test Series

Exam Paper Generator

Online Exam Module