Trigonometric Ratios of Some Specific Angles

If $2\cos \left(\theta \right)-\sin \left(\theta \right)=\frac{1}{\sqrt{2}}$ and $\left(0°<\theta <90°\right)$, then the value of $2\sin \left(\theta \right)+\cos \left(\theta \right)$ is:

If $\tan \left(\frac{\mathrm{\pi }}{2}-\frac{\theta }{2}\right)=\sqrt{3}$, then value of $\cos \left(\theta \right)$ is:

If $\mathrm{sin\theta }+\mathrm{cosec\theta }=2$, then what is the value of ${\sin }^2\mathrm{\theta }+{\mathrm{cosec}}^5\mathrm{\theta }$ when $0°\le \theta \le 90°$?

Find the value $3\sin \left(20°\right)-4{\sin }^3\left(20°\right)$:

If $\tan \theta -\cot \theta =0,$ then the value of $\sin \theta +\cos \theta ?$

If $\sin 45°-x\cos 60°=0$, then what is the value of $x$?

If $\sin \theta =\frac{\sqrt{3}}{2}$, show that $4{\cos }^3\theta -3\cos \theta =-1$

In an acute angled triangle $ABC$, if $\sin (A+B-C)=\frac{1}{2}$ and $\cos \left(B+C-A\right)=\frac{1}{2}$, then find $\angle A, \angle B$ and $\angle C$. 

If $\sin \theta =\frac{3}{5}$ and $\cos \theta =\frac{4}{5}$, then find the value of ${\sin }^2\theta +{\cos }^2\theta$.

If $4{\cot }^{2}45°-{\sec }^{2}60°+{\sin }^{2}60°+p=\frac{3}{4}$, then find the value of $p$

A tower and a building are standing vertically on a level ground. The angles of elevation of the top of the tower from a point on the same ground and from the top of the building are found to be $30° \text{and} 60°$ respectively. If the distance of the point from the foot of the tower is $30\sqrt{3} \mathrm{m}$ and height of the building is $10 \mathrm{m}$, then find the distance between the foot of the tower and building and also the distance between their tops.

In a rectangle $ABCD$, the length $BC$is twice the width $AB$. Pick a point $P$ on side $BC$ such that the lengths of $AP$ and $BC$ are equal. The measure of angle$\angle CPD$ is:

What is the value of $\sin 30°$?

$\cot 30° + \tan 60°$

Find the value of $5\sin 30°+3\tan 45°$

Given that $\tan \left({\theta }_1+{\theta }_2\right)=\frac{\tan {\theta }_1+\tan {\theta }_2}{1-\tan {\theta }_1\tan {\theta }_2}.$

Find $\left({\theta }_1+{\theta }_2\right)$ when $\tan {\theta }_1=\frac{1}{2}, \tan {\theta }_2=\frac{1}{3}$

$\mathrm{tan\theta }=\mathrm{cot\theta }$ for all values of $\theta$.

The angle of elevation of the top of a rock from the top and foot of a $100 \mathrm{m}$ high tower are $30°\text{ and }45°$ respectively. Find the height of the rock.

If $\tan A+{\sin }^2 45°-{\cos }^2 30°-{\mathrm{cosec}}^2 45°{\sec }^2 60°= 4$, then find $\tan A$.

Prove that $\sqrt{6}.(\sqrt{2}+\sqrt{3})[\frac{\mathrm{cosec}60°}{2}-\cos 45°]=-1$.