Trigonometric Ratios and Relation Between Them

If $\cos \theta =\frac{b}{\sqrt{a^2+b^2}}$ then prove that ${\left(\frac{b+\sqrt{a^2+b^2}}{a}\right)}^2=\frac{\sqrt{a^2+b^2}+b}{\sqrt{a^2+b^2}-b}=\frac{1+\cos \theta }{1-\cos \theta }$

In $∆PQR$, $\angle Q$ is a right angle. If $PQ=3 \mathrm{cm}$ and  $PR=6 \mathrm{cm}$, then the measurement of $\angle QPR$ is:

If $\sin \theta = \frac{5}{13} \text{and} AC= 52 \mathrm{cm}$, find $AB, BC \text{and} \tan \theta$.

From the following figures, find the values of $\sin A, \sec A$ and ${\cos }^2 A+{\sin }^2 A$.

Prove: ${\left(\tan \theta +cot \theta \right)}^2-{\left(\tan \theta -cot \theta \right)}^2=4$

Prove: 

$\frac{\tan x+\sin x}{\tan x-\sin x}=\frac{\sec x+1}{\sec x-1}$

Explain why ${\sin }^{-1}\theta \ne \frac{1}{\sin \theta }$, but ${\sin }^2\theta ={\left(\sin \theta \right)}^2$.

$\mathrm{cosec}A=\frac{1}{\sin B}$.

if $\tan \theta =\frac{5}{12}$ find the value of $\frac{3 \cos \theta +5\sin \theta }{3 \cos \theta -5\sin \theta }$

if $\sin A+\mathrm{cosec} A=2$ find the value of ${\sin }^2\mathrm{A}+{\mathrm{cosec}}^2\mathrm{A}$

Use Fig. to find the value of $3 \tan x^0-2 \sin y^0 + 4 \cos y^0$ 

Given $\text{5 cos A-12 sin A = 0}$ find the value of$\frac{\sin \mathrm{A} + \cos \mathrm{A} }{2 \cos \mathrm{A} - \sin \mathrm{A}}$ 

if $5 \tan \theta = 4,$find the value of $\frac{5 \sin \theta - 3 \cos \theta }{5 \sin \theta + 2 \cos \theta }$

In Fig ABC is a right triangle, righrt angled at B. Given that $\angle \mathrm{ACB}=\mathrm{\theta }$ side $\mathrm{AB}=2 \mathrm{cm}$ and side $\mathrm{BC}=1 \mathrm{cm}$, find the value of  ${\sin }^2\theta +{\cos }^2\theta$.

If $2 \sin x = \sqrt{3}$, then evaluate $4 {\sin }^{3}x - 3 \sin x$

if $\frac{\sec \theta - \tan \theta }{\sec \theta + \tan \theta }=\frac{1}{4}$ find $\sin \theta$

In a $△\mathrm{ABC},$$\mathrm{AD}\perp \mathrm{BC}$ if $\tan \mathrm{B} =\frac{3}{4}$ and $\tan \mathrm{C} = \frac{5}{12}$ and $\mathrm{BC} = 56 \mathrm{cm}$ calculate the lengths of $\mathrm{AD}$  

Find the trigonometric ratios in standard position whose terminal arm passes through the points $\left(-6, 8\right)$

If $\cos \theta =\frac{-7}{25}$ and $\theta$ is in second quadrant, find the remaining trigonometric ratios.

Find the sine ratio in standard position where terminal arm passes through $\left(-5,-12\right)$.