Trigonometry Questions

A tower stands at the centre of a circular park. $A$ and $B$ are two points on the boundary of the park such that $AB(=a)$ subtends an angle of $60^{\circ }$ at the foot of the tower, and the angle of elevation of the top of the tower from $A$ or $B$ is $30^{\circ }$. The height of the tower is:

If ${\sin }^{-1}\left(\frac{x}{5}\right)+{\text{cosec}}^{-1}\left(\frac{5}{4}\right)=\frac{\pi }{2}$, then a value of $x$ is:

If $0<x<\pi$ and $\cos x+\sin x=1/2$, then $\tan x$ is:

A particle has two velocities of equal magnitude $u$ inclined to each other at an angle $\theta$. If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then $\theta$ is:

The value of ${\sum }_{k=1}^{10}\left(\sin \frac{2k\pi }{11}+i\cos \frac{2k\pi }{11}\right)$ is:

The number of values of $x$ in the interval $[0,3\pi ]$ satisfying the equation $2{\sin }^{2}x+5\sin x-3=0$ is:

If the roots of the quadratic equation $x^2+px+q=0$ are $\tan 30^{\circ }$ and $\tan 15^{\circ }$, respectively, then the value of $2+q-p$ is:

In a $\Delta PQR$, if $3\sin P+4\cos Q=6$ and $4\sin Q+3\cos P=1$, then the angle $R$ is equal to:

In a triangle $PQR$, $\angle R=\frac{\pi }{2}$. If $\tan \frac{P}{2}$ and $\tan \frac{Q}{2}$ are the roots of $a{x}^2+bx+c=0,a\ne 0$, then:

If $A={\sin }^{2}x+{\cos }^{4}x$, then for all real $x$:

Let $\cos (\alpha +\beta )=\frac{4}{5}$ and let $\sin (\alpha -\beta )=\frac{5}{13}$, where $0\le \alpha ,\beta \le \frac{\pi }{4}$, then $\tan 2\alpha =$

Let $A$ and $B$ denote the statements:$A:\cos \alpha +\cos \beta +\cos \gamma =0$$B:\sin \alpha +\sin \beta +\sin \gamma =0$ If $\cos (\beta -\gamma )+\cos (\gamma -\alpha )+\cos (\alpha -\beta )=-\frac{3}{2}$, then:

$AB$ 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 $60^{\circ }$. He moves away from the pole along the line $BC$ to a point $D$ such that $CD=7\text{m}$. From $D$, the angle of elevation of the point $A$ is $45^{\circ }$. Then the height of the pole is:

The upper $\frac{3}{4}$ th portion of a vertical pole subtends an angle ${\tan }^{-1}\frac{3}{5}$ at point in the horizontal plane through its foot and at a distance 40 m from the foot. A possible height of the vertical pole is:

If in a triangle $ABC$, $a{\cos }^2\left(\frac{C}{2}\right)+c{\cos }^2\left(\frac{A}{2}\right)=\frac{3b}{2}$, then the sides $a,b$ and $c$:

The sum of the radii of inscribed and circumscribed circles for an $n$ -sided regular polygon of side $a$, is:

If in a triangle $ABC$, the altitudes from the vertices $A,B,C$ on opposite sides are in H.P., then $\sin A,\sin B,\sin C$ are in:

If ${\cos }^{-1}x-{\cos }^{-1}\frac{y}{2}=\alpha$, then $4x^2-4xy\cos \alpha +y^2$ is equal to:

In a triangle $ABC$, let $\angle C=\frac{\pi }{2}$. If $r$ is the inradius and $R$ is the circumradius of the triangle, then $2(r+R)$ equals:

If $f:R\to S,$ defined by $f(x)=\sin x-\sqrt{3}\cos x+1,$ is onto, then the interval of $S$ is

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is $60^{\circ }$ and when he retires $40$ meter away from the tree the angle of elevation becomes $30^{\circ }.$ The breadth of the river is

The sides of a triangle are $\sin \alpha ,\cos \alpha$ and $\sqrt{1+\sin \alpha \cos \alpha }$ for some $0<\alpha <\frac{\pi }{2}.$ Then the greatest angle of the triangle is

If $u=\sqrt{a^2{\cos }^2\theta +b^2{\sin }^2\theta }+\sqrt{a^2{\sin }^2\theta +b^2{\cos }^2\theta },$ then the difference between the maximum and minimum values of $u^2$ is given by

Let $\alpha ,\beta$ be such that $\pi <\alpha -\beta <3\pi .$ If $\sin \alpha +\sin \beta =-\frac{21}{65}$ and $\cos \alpha +\cos \beta =-\frac{27}{65},$ then the value of $\cos \frac{\alpha -\beta }{2}$ is