Basics of Trigonometry

If $\tan \theta =-\frac{4}{3},$ then $\sin \theta$ is –

$\sum _{k=1}^3{\cos }^2\left(2k-1\right)\frac{\mathrm{\pi }}{2}=$

If $\tan \theta =\frac{5}{12}$; then what is the value of $3\sin \theta +2\cos \theta$.

If $\frac{\sin A}{1+\cos A}+\frac{1+\sin A}{\cos A}=\frac{\left(\sin B+\cos B\right)\left(1+\sin 2A\right)-\left[\cos \left(A+B\right)+\sin \left(A-B\right)\right]}{\sin \left(A+B\right)+{\cos }^{2}A\sin B-{\sin }^{2}A\cos B}$, then -

How do you convert 1 min in radian ?

An arc of certain length subtends ${20}^0{17}^{\text{'}}$ at the centre of a circle of radius $6$ cm. Find in sexagesimal unit the angle subtended by the same arc at the centre of a circle of radius $8$ cm.

If $0°<\theta <90°$ and $(\sin \theta +cosec\theta )=\frac{5}{2}$, then $\theta =?$

Evaluate the following $2{\tan }^{2}45°+{\cos }^{2}30°-{\sin }^{2}60°$.

Two parallel chords are drawn on the same side of the centre of a circle of radius $20.$ It is found that they subtend $60°$ and $120°$ angles at the centre of the circle, then the perpendicular distance between the chords is

If ${2021}^{{\left({\log }_{2021}2022\right)}^x}={2022}^{{\left({\log }_{2022}2021\right)}^x}$ and $x=\frac{a}{b}$, where $a$ & $b$ are co-prime numbers, then find the value of $\frac{1}{2}\sin \left(\frac{a\mathrm{\pi }}{3b}\right)$ equals

Find the value of $cosec750°-2\left(\cot 765°\right)$

The radius of the circle in which a central angle of $\frac{\mathrm{\pi }}{3}$radian intercepts an arc of length $22\mathrm{cm}$ is 

If value of $\frac{3\cos {36}^{°}+5\sin {18}^{°}}{5\cos {36}^{°}-3\sin {18}^{°}}$ is $\frac{a\sqrt{5}-b}{c}$ where $a,b,c$ are natural numbers and $\gcd \left(a,c\right)=1$ then $a+b+c$ is

If $F\left(k\right)=\left(1+\mathrm{s}\mathrm{i}\mathrm{n}\frac{\pi }{2k}\right)\left(1+\sin \left(k-1\right)\frac{\pi }{2k}\right)$$\left(1+\sin \left(2k+1\right)\frac{\pi }{2k}\right)\left(1+\sin \left(3k-1\right)\frac{\pi }{2k}\right)$, then the value of $F\left(1\right)+F\left(2\right)+F\left(3\right)$ is equal to

If $\tan A=1$ and $\tan B=\sqrt{3},$ then $\cos (A+B)$ is equal to

If $\tan \left(A-B\right)=\frac{1}{\sqrt{3}}$ and $\tan \left(A+B\right)=\sqrt{3}$ then the values of $A$ and $B$ are respectively

If ${\tan }^2\theta +\sec 2\theta =1$ then $\tan \theta$ can be:

The number of solutions of $2{\cos }^2\left(\frac{x}{2}\right){\sin }^{2}x=x^2+\frac{1}{x^2},0\le x\le \frac{\pi }{2}$ is:

The maximum value of $y=\frac{{\sec }^2\theta -\tan \theta }{{\sec }^2\theta +\tan \theta }$ , then

The length of each side of a regular dodecagon is 20 cm then radius of its circumscribing circle is