Let in a right-angled triangle,the smallest angle be $\theta$. If a triangle formed by taking the reciprocal of its sides is also a right-angled triangle,then $\sin \theta$ is equal to:

In a triangle $ABC$ with a fixed base $BC$,the vertex $A$ moves such that $\cos B + \cos C = 4 \sin^2 \frac{A}{2}$. If $a, b,$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A, B,$ and $C$,respectively,then:
$(A) b+c=4a$
$(B) b+c=2a$
$(C) \text{locus of point } A \text{ is an ellipse}$
$(D) \text{locus of point } A \text{ is a pair of straight lines}$

In $\triangle ABC$,find the value of $a^3 \cos(B-C) + b^3 \cos(C-A) + c^3 \cos(A-B)$.

In a triangle $ABC$,$AD$ is the altitude from $A$. Given $b > c$,$\angle C = 23^o$ and $AD = \frac{abc}{b^2 - c^2}$,then $\angle B = \dots^o$.

The angles $A, B$ and $C$ of a triangle $ABC$ are in $A.P.$ and $a : b = 1 : \sqrt{3}$. If $c = 4 \text{ cm}$,then the area (in $\text{sq. cm}$) of this triangle is:

In $\Delta ABC$,${a^2}({\cos ^2}B - {\cos ^2}C) + {b^2}({\cos ^2}C - {\cos ^2}A) + {c^2}({\cos ^2}A - {\cos ^2}B) = $