Answer
Verified
412.2k+ views
Hint: Here we can proceed by finding the $\sin $ of the same angle as given and we know that $\csc x = \dfrac{1}{{\sin x}}$ and therefore we can divide both the values of the $\sin $ of the same angle and get the exact value of the $\csc \left( {\dfrac{\pi }{6}} \right)$.
Complete step by step solution:
Now we are given to find the exact value of $\csc \left( {\dfrac{\pi }{6}} \right)$
We know that:
$\sin \left( {\dfrac{\pi }{6}} \right) = \dfrac{1}{2}$$ - - - - (1)$
Now we can find the relation between $\sin ,\csc $ to get the value of the $\csc \left( {\dfrac{\pi }{6}} \right)$
Let us consider the triangle $ABC$ right-angled at $B$
We know that:
$\sin \theta = \dfrac{{{\text{perpendicular}}}}{{{\text{hypotenuse}}}} - - - - (2)$
We also know that:
$\csc \theta = \dfrac{{{\text{hypotenuse}}}}{{{\text{perpendicular}}}} - - - - (3)$
Now if we multiply the equation (2) and (3) we will get:
\[\sin \theta .\csc \theta = \dfrac{{{\text{perpendicular}}}}{{{\text{hypotenuse}}}} \times \dfrac{{{\text{hypotenuse}}}}{{{\text{perpendicular}}}} = 1\]
Hence we get that:
\[\sin \theta .\csc \theta = 1\]$ - - - (4)$
Now substituting the value of $\sin \left( {\dfrac{\pi }{6}} \right) = \dfrac{1}{2}$ we got in equation (1) in the above equation (4), we get:
\[\sin \theta .\csc \theta = 1\]
\[
\sin \dfrac{\pi }{6}.\csc \dfrac{\pi }{6} = 1 \\
\dfrac{1}{2}.\csc \dfrac{\pi }{6} = 1 \\
\]
So we know the value of $\sin \left( {\dfrac{\pi }{6}} \right) = \dfrac{1}{2}$
So putting it in above, we get:
\[
\dfrac{1}{2}.\csc \dfrac{\pi }{6} = 1 \\
\csc \dfrac{\pi }{6} = 2 \\
\]
Hence for this, we must know all the trigonometric relations between all trigonometric functions because due to this all the general values of all trigonometric functions can be found.
Note:
Here in these types of problems where we are asked to find the value of the tangent or cotangent of any angle, we must know the basic values of the sine and cosine of the angles like $0^\circ,30^\circ,45^\circ,60^\circ,90^\circ $ and then we can easily calculate the same angles of the tangent, cotangent, secant, and cosecant of that same angle.
Complete step by step solution:
Now we are given to find the exact value of $\csc \left( {\dfrac{\pi }{6}} \right)$
We know that:
$\sin \left( {\dfrac{\pi }{6}} \right) = \dfrac{1}{2}$$ - - - - (1)$
Now we can find the relation between $\sin ,\csc $ to get the value of the $\csc \left( {\dfrac{\pi }{6}} \right)$
Let us consider the triangle $ABC$ right-angled at $B$
We know that:
$\sin \theta = \dfrac{{{\text{perpendicular}}}}{{{\text{hypotenuse}}}} - - - - (2)$
We also know that:
$\csc \theta = \dfrac{{{\text{hypotenuse}}}}{{{\text{perpendicular}}}} - - - - (3)$
Now if we multiply the equation (2) and (3) we will get:
\[\sin \theta .\csc \theta = \dfrac{{{\text{perpendicular}}}}{{{\text{hypotenuse}}}} \times \dfrac{{{\text{hypotenuse}}}}{{{\text{perpendicular}}}} = 1\]
Hence we get that:
\[\sin \theta .\csc \theta = 1\]$ - - - (4)$
Now substituting the value of $\sin \left( {\dfrac{\pi }{6}} \right) = \dfrac{1}{2}$ we got in equation (1) in the above equation (4), we get:
\[\sin \theta .\csc \theta = 1\]
\[
\sin \dfrac{\pi }{6}.\csc \dfrac{\pi }{6} = 1 \\
\dfrac{1}{2}.\csc \dfrac{\pi }{6} = 1 \\
\]
So we know the value of $\sin \left( {\dfrac{\pi }{6}} \right) = \dfrac{1}{2}$
So putting it in above, we get:
\[
\dfrac{1}{2}.\csc \dfrac{\pi }{6} = 1 \\
\csc \dfrac{\pi }{6} = 2 \\
\]
Hence for this, we must know all the trigonometric relations between all trigonometric functions because due to this all the general values of all trigonometric functions can be found.
Note:
Here in these types of problems where we are asked to find the value of the tangent or cotangent of any angle, we must know the basic values of the sine and cosine of the angles like $0^\circ,30^\circ,45^\circ,60^\circ,90^\circ $ and then we can easily calculate the same angles of the tangent, cotangent, secant, and cosecant of that same angle.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE