Exercise 10 (a)

In a $△ABC$ if $C=90°$, then prove that $\left(\frac{a^2+b^2}{a^2-b^2}\right)\sin \left(A-B\right)=1$.

In a $△ABC$ show that $a^2\sin 2C+c^2\sin 2A=4\Delta$.

A lamp post is situated at the middle point $M$ of the side $AC$ of a triangular plot $ABC$ with $BC=7\mathrm{m}$, $CA=8\mathrm{m}$ and $AB=9\mathrm{m}$. Lamp post subtends an angle $15°$ at the point $B$. Find the height of the lamp post.

Two ships leave a port at the same time. One goes $24 \mathrm{km}$ per hour in the direction $N45°E$ and other travels $32 \mathrm{km}$ per hour in the direction $S75°E$. Find the distance between the ships at the end of $3$ hours.

A tree stands vertically on the slant of the hill. From a point $A$ on the ground $35$ meters down the hill from the base of the tree, the angle of elevation of the top of the tree is $60°$. If the angle of elevation of the foot of the tree from $A$ is $15°$, then find the height of the tree.

The upper ${\frac{3}{4}}^{\mathrm{th}}$ portion of a vertical pole subtends an angle ${\tan }^{-1}\frac{3}{5}$ at a point in the horizontal plane through its foot and at a distance $40\mathrm{m}$ from the foot. Given that the vertical pole is at a height less than $100\mathrm{m}$ from the ground, find its height.

$AB$ is a vertical pole with $B$ at the ground level and $A$ at the top. A man finds that the angle of elevation of the point $A$ from a certain point $C$ on the ground is $60°$. He moves away from the pole along the line $BC$ to a point $D$ such that $CD=7\mathrm{m}$. From $D$, the angle of elevation of the point $A$ is $45°$. Find the height of the pole.

Let an object be placed at some height $h\mathrm{cm}$ and let $P$ and $Q$ be two points of observation which are at a distance $10\mathrm{cm}$ apart on a line inclined at angle $15°$ to the horizontal. If the angles of elevation of the object from $P$ and $Q$ are $30°$ and $60°$ respectively, then find $h$.