Question

If ${\text{x= tan 4}}{{\text{5}}^0}.{\text{ cot 6}}{{\text{0}}^0} = \sin {\text{ 3}}{{\text{0}}^0}{\text{cosec 6}}{{\text{0}}^0}$, then the value of ${\text{x}}$ is
A) ${\text{1}}$
B) $\dfrac{1}{4}$
C) $\dfrac{1}{2}$
D) $\sqrt 3 $

Answer 1

Hint: In this question we have to find the value of ${\text{x}}$. For that we are going to find the value by using trigonometric table values. The three basic functions in trigonometry are sine, cosine and tangent. The trigonometric table is useful in the number of areas. And also we are going to calculate the value and it has been given in a complete step by step solution.

Formula used: $\tan {45^0} = 1$
$\cot {\text{6}}{{\text{0}}^0} = \dfrac{1}{{\sqrt 3 }}$
$\sin {\text{ 3}}{{\text{0}}^0} = \dfrac{1}{2}$
${\text{cosec 6}}{{\text{0}}^0} = {\text{ }}\dfrac{2}{{\sqrt 3 }}$

Complete step-by-step answer:
Trigonometric ratios table helps to find the value of trigonometric standard angles such as $0,{\text{ 30, 45, 60}}$ and $90$.
It consists of trigonometric series ratios -sine, cosine, tangent, cosecant, secant, and cotangent.
Given that ${\text{x tan 4}}{{\text{5}}^0}.{\text{ cot 6}}{{\text{0}}^0} = \sin {\text{ 3}}{{\text{0}}^0}{\text{cosec 6}}{{\text{0}}^0}$
To find the value of ${\text{x}}$.
Now, ${\text{x tan 4}}{{\text{5}}^0}.{\text{ cot 6}}{{\text{0}}^0} = \sin {\text{ 3}}{{\text{0}}^0}{\text{cosec 6}}{{\text{0}}^0}$
Here substitute the trigonometric values,
We know that trigonometric standard angles such as $0,{\text{ 30, 45, 60}}$ and $90$.
That implies,
\[\left( {\text{x}} \right)\left( 1 \right)\left( {\dfrac{1}{{\sqrt 3 }}} \right) = \left( {\dfrac{1}{2}} \right)\left( {\dfrac{2}{{\sqrt 3 }}} \right)\]
By cancelling the terms, we get the value
${\text{x = 1}}$
Hence the value of ${\text{x}}$ is $1$.

$\therefore $ Option A is the correct answer.

Note: In this type of question we have to find the value of x. For that we used trigonometric standard angles values.
By substituting the value, we get the final answer.
Many geometric calculations can be easily figured out using the table of trigonometric functions and formulas as well.
The trigonometric table is useful in the number of areas. It is essential for navigation, science and engineering.
The table was effectively used in the pre digital era, even before the existence of pocket calculators. Further, the table led to the development of the first mechanical computing devices.