Question

If $\cot \theta = 2$ , then find the value of all T – ratios of $\theta$?

Answer 1

Hint: First, you need to consider a right angled triangle. Since $\cot \theta = 2$, we get the value of the base of the triangle as 2 and the value of the perpendicular of the triangle as 1. Now you need to find the hypotenuse. And from all the values you need to find the T- ratios. You should use the formulas $\sin \theta = \dfrac{perpendicular}{hypotenuse}$, $\cos \theta = \dfrac{base}{hypotenuse}$, $\tan \theta = \dfrac{perpendicular}{base}$, $\csc \theta = \dfrac{1}{\sin \theta}$, $\sec \theta = \dfrac{1}{\cos \theta}$. Therefore, using these formulas, you get all the values of T - ratios.

Complete step by step solution:
Here is the step wise solution.
The first step to do is to find the value of base and the perpendicular of a right angled triangle from the equation $\cot \theta = 2$. Therefore, we get the value of the base of the triangle as 2 and the value of the perpendicular of the triangle as 1.
Next step is to find the hypotenuse. So we use the formula
$\Rightarrow base^2 + perpendicular^2 = hypotenuse^2$
$\Rightarrow hypotenuse^2 = 2^2 + 1 = 5$
$\Rightarrow hypotenuse = \sqrt{5}$

Now we use the trigonometric identities to find all the values.
$\sin \theta = \dfrac{perpendicular}{hypotenuse} = \dfrac{1}{\sqrt{5}}$
 $\cos \theta = \dfrac{base}{hypotenuse} = \dfrac{2}{\sqrt{5}}$
$\tan \theta = \dfrac{perpendicular}{base} = \dfrac{1}{2}$
$\csc \theta = \dfrac{1}{\sin \theta} = \sqrt{5}$
 $\sec \theta = \dfrac{1}{\cos \theta} = \dfrac{\sqrt{5}}{2}$.
Therefore, as we can see , we get the final answer for the question as
$\sin \theta = \dfrac{1}{\sqrt{5}}$, $\cos \theta = \dfrac{2}{\sqrt{5}}$, $\tan \theta = \dfrac{1}{2}$, $\csc \theta = \sqrt{5}$, $\sec \theta = \dfrac{\sqrt{5}}{2}$.

Note: You need to remember all the trigonometric identities properly. They are very helpful in these problems. To remember them, you can solve many problems on them. So you will be able to learn and remember them properly. Also you need to learn all the relations of the trigonometric identities to a triangle. You need to be careful not to confuse between the formulas for sin and cos.