If in a $∆ABC, \tan A+\tan B+\tan C>0,$ then the triangle is

Important Questions on Trigonometric Ratios and Identities

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

If $\mathrm{s}\mathrm{i}{\mathrm{n}}^4\alpha +4\mathrm{c}\mathrm{o}{\mathrm{s}}^4\beta +2=4\sqrt{2}\mathrm{s}\mathrm{i}\mathrm{n}\alpha \mathrm{c}\mathrm{o}\mathrm{s}\beta$, $\alpha , \beta \in \left[0,\mathrm{\pi }\right]$, then $\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha +\beta \right)-\mathrm{c}\mathrm{o}\mathrm{s}\left(\alpha -\beta \right)$ is equal to

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

The maximum value of $3\cos \theta +5\sin \left(\theta -\frac{\pi }{6}\right)$ for any real value of $\theta$ is :

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

If the angle of elevation of a cloud from a point $P$ which is $25\mathrm{m}$ above a lake be $30°$ and the angle of depression of reflection of the could in the lake from $P$ be $60°$, then the height of the cloud (in meters) from the surface of the lake is :

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

If $\cos \alpha +\cos \beta =\frac{3}{2} \text{and} \sin \alpha +\sin \beta =\frac{1}{2}$and $\theta$ is the arithmetic mean of $\alpha \& \beta ,$ then $\sin 2\theta +\cos 2\theta$ is equal to:

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

In $\Delta ABC$, if $\tan A+\tan B+\tan C=6$ and $\tan A\tan B=2$ then  $\tan C=$ ……

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

$ABCD$ is a trapezium such that $AB$ and $CD$ are parallel and $BC\perp CD$. If $\angle ADB=\theta$, $BC=p$ and $CD=q$, then $AB$ is equal to

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

The value of $\cos \frac{\pi }{2^2}\cdot \cos \frac{\pi }{2^3}\cdot \dots \cdot \cos \frac{\pi }{2^{10}}\cdot \sin \frac{\pi }{2^{10}}$ is:

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

If $x{\sin }^3\theta +y{\cos }^3\theta =\sin \theta \cos \theta$ and $x\sin \theta =y\cos \theta$, then $x^2+y^2$ is

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

In a triangle $ABC,{\left(\tan \frac{A}{2}\tan \frac{B}{2}\tan \frac{C}{2}\right)}^2\le$

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

The value of $\sum _{k=1}^{13}\frac{1}{\sin \left(\frac{\pi }{4}+\frac{\left(k-1\right)\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{k\pi }{6}\right)}$ is equal to

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

In a $\mathrm{\Delta }ABC$, $\frac{a}{b}=2+\sqrt{3}$, and $\angle C=60°.$ Then the ordered pair $(\angle A,\angle B)$ is equal to:

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

If $A, B, C$ are the angle of $\mathrm{\Delta }ABC$ then $\cot A\cot B+\cot B\cot C+\cot A\cot C=$

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

A bird is sitting on the top of a vertical pole $20 \mathrm{m}$ high and its elevation from a point $O$ on the ground is $45°$. It flies off horizontally straight away from the point $O$. After one second, the elevation of the bird from $O$ is reduced to $30°$. Then the speed (in $\mathrm{m}/\mathrm{s}$) of the bird is 

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

The expression $\frac{\mathrm{tanA}}{1-\mathrm{cotA}}+\frac{\mathrm{cotA}}{1-\mathrm{tanA}}$ can be written as :

HARD

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

The integer part of the number $\sum _{\mathrm{k}=0}^{44}\frac{1}{\mathrm{cosk}°\cos \left(\mathrm{k}+1\right)°}$ is $\left(\sin 1°=0.0174524\right)$

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

If an angle $A$ of a $\mathrm{\Delta }ABC$ satisfies $5\cos A+3=0$, then the roots of the quadratic equation $9x^2+27x+20=0$ are

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

The value of $\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{14\pi }{15}$ is equal to

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

Consider a triangular plot $ABC$ with sides $AB=7 m,$ $BC=5 m$ and $CA=6 m.$ A vertical lamp-post at the mid-point $D$ of $AC$ subtends an angle $30°$ at $B.$ The height (in $m$) of the lamp-post is:

EASY

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

In $\Delta ABC;$ with usual notations, $\frac{b\mathrm{s}\mathrm{i}\mathrm{n}B-c\mathrm{s}\mathrm{i}\mathrm{n}C}{\sin \left(B-C\right)}$ = ……

MEDIUM

Mathematics>Trigonometry>Trigonometric Ratios and Identities>Conditional Identities

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 $\cos \left(\alpha +\frac{\pi }{2}\right),\cos \left(\beta +\frac{\pi }{2}\right)$ and $\cos \left(\gamma +\frac{\pi }{2}\right)$ is equal to