Question

If $\cos ec\theta =\sqrt{10}$ then find the value of other trigonometric ratios for $\theta $.

Answer 1

Hint:Here we may use the trigonometric formulas to calculate the base, hypotenuse and height of the right angled triangle for which $\cos ec\theta =\sqrt{10}$. So, using these values we may find the value of all the other trigonometric ratios for $\theta $.

Complete step-by-step answer:
Since, we know that cosecant of a triangle is the ratio of hypotenuse of the triangle to the perpendicular of the triangle. So, according to this we have:
$\cos ec\theta =\dfrac{hypotenuse}{perpendicular}.........(1)$
Now we may substitute the given value of $\cos ec\theta =\sqrt{10}$ in equation (1), and on substitution we get:
$\sqrt{10}=\dfrac{hypotenuse}{perpendicular}$
On comparing both sides of this equation we get:
Length of hypotenuse of the triangle = $\sqrt{10}$ units
And, length of perpendicular of the triangle = 1 unit
Now, we may use Pythagoras theorem to find the length of base of this triangle. So, according to Pythagoras theorem we have:
${{\left( hypotenuse \right)}^{2}}={{\left( perpendicular \right)}^{2}}+{{\left( base \right)}^{2}}..........(2)$
On substituting the value of hypotenuse and perpendicular, we get:
$\begin{align}
  & {{\left( \sqrt{10} \right)}^{2}}={{\left( 1 \right)}^{2}}+{{\left( base \right)}^{2}} \\
 & 10=1+{{\left( base \right)}^{2}} \\
 & base=\sqrt{10-1} \\
 & base=\sqrt{9} \\
 & base=3\,\,units \\
\end{align}$
 So, the length of the base of this triangle is = 10 units.
Using the values of hypotenuse, perpendicular and base, we can now get the values of other trigonometric ratios as:
$\sin \theta =\dfrac{perpendicular}{hypotenuse}=\dfrac{1}{\sqrt{10}}$
$\cos \theta =\dfrac{base}{hypotenuse}=\dfrac{3}{\sqrt{10}}$
$\tan \theta =\dfrac{perpendicular}{base}=\dfrac{1}{3}$
Since, we know that sec and cosec are reciprocal to each other. So, we have:
$\sec \theta =\dfrac{1}{\cos \theta }=\dfrac{\sqrt{10}}{3}$
Also, we know that cot and tan are reciprocal to each other. So, we have:
$\cot \theta =\dfrac{1}{\tan \theta }=3$
Hence, the values of the other trigonometric ratios corresponding to $\cos ec\theta =\sqrt{10}$ are:
$\sin \theta =\dfrac{1}{\sqrt{10}}$,$\cos \theta =\dfrac{3}{\sqrt{10}}$,$\tan \theta =\dfrac{1}{3}$,$\sec \theta =\dfrac{\sqrt{10}}{3}$ and $\cot \theta =3$.

Note: Students should note here that we can also find the values of all other trigonometric ratios using trigonometric formulas and identities relating various ratios.