Question

Use $\tan \theta = 4$ to find $\cos \theta$

Answer 1

Hint : To solve this problem, we should know the formula for $\tan \theta$ in the form of a right-angled triangle. With this, use the Pythagoras theorem to solve this problem. Don’t forget to consider the number $4$ as a ratio.

Complete step-by-step answer :
The given question is,
$\tan \theta = 4$
We know that the formula for,
$\tan \theta = \dfrac{{opposite}}{{adjacent}}$ $= \dfrac{4}{1}$ , This is the ratio of $4$ and $1$ , but not the exact value and hence we take the value for $opposite$ side as $4x$ and the value for $adjacent$ side as $x$ . To find the value for $hypotenuse$ side we consider Pythagora's theorem, which states that the square of the hypotenuse side is equal to the sum of squares of the other two sides. If we apply this theorem we get,
$hy{p^2} = op{p^2} + ad{j^2}$
This theorem can be also written as,
$hyp = \sqrt {op{p^2} + ad{j^2}}$ … (1)
As we know that the ratio for the $opposite$ and $adjacent$ side, let’s substitute the values in the equation (1) we get,
$hyp = \sqrt {4{x^2} + {x^2}} \\\ hyp = \sqrt {5{x^2}} \\\ hyp = \sqrt 5 x \;$ $$
We got the value for $hypotenuse$ side in term of $x$ and now let’s consider the formula for $\cos \theta$ which is equal to,
$\cos \theta = \dfrac{{adj}}{{hyp}}$
Now we know the value of $adj$ and $hyp$ is equal to $x$ and $\sqrt 5 x$ . When we substitute these values in $\cos \theta$ we get,
$\cos \theta = \dfrac{x}{{\sqrt 5 x}} \\\ \cos \theta = \dfrac{1}{{\sqrt 5 }} \;$
This is the required solution.
So, the correct answer is “ $\cos \theta = \dfrac{1}{{\sqrt 5 }}$ ”.

Note : The very beautiful part of ratio is, with this we can find the values in a simple way if one hint is provided in the question. Don’t forget that whenever there exists a ratio, you should view it as if It was a very big hint.
Only when you consider a right-angled triangle, you should use the Pythagoras theorem. And when we considered $\tan \theta$ , we had the value for $opposite$ and $adjacent$ side and in order to find the value of the $hypotenuse$ side, we used the Pythagoras theorem.