Question

If \[\sin\ \theta = \dfrac{4}{7}\].what is \[\cos\ \theta\] ?

Answer 1

Hint:In this question, given that \[\sin\ \theta = \dfrac{4}{7}\] then, we need to find the value of \[\cos\ \theta\] . By using trigonometric identities , we can find the \[\cos\ \theta\] . First we can use Pythagoras trigonometric identity \[\ \cos^{2}\theta + \sin^{2}\theta = 1\]. Then we need to subtract \[\sin^{2}\theta\] on both sides. Then on taking square root and simplifying, we get the expression in the form of \[\cos\ \theta\] and \[\sin\ \theta\]. Then we need to substitute the value of \[\sin\ \theta\] and on further simplifying, we can find the value of \[\cos\ \theta\] .

Complete step by step answer:
Given, \[\sin\ \theta = \dfrac{4}{7}\]
Here we need to find \[\cos\ \theta\] .
By using Pythagoras trigonometric identity,
\[\Rightarrow \cos^{2}\theta + \sin^{2}\theta = 1\]
On subtracting \[\sin^{2}\theta\] on both sides,
We get,
\[\Rightarrow \cos^{2}\theta + \sin^{2}\theta - \sin^{2}\theta = 1\ - \sin^{2}\theta\]
On simplifying ,
We get,
\[\Rightarrow \cos^{2}\theta = 1 - \sin^{2}\theta\]
On taking square root on both sides,
\[\Rightarrow \ \cos\ \theta = \pm \sqrt{1 - \sin^{2}\theta}\]
Now on substituting the value of \[\sin\ \theta\] ,
We get,
\[\Rightarrow \cos\ \theta = \pm \sqrt{1 - \left( \dfrac{4}{7} \right)^{2}}\]
On simplifying,
We get,
\[\Rightarrow \cos\ \theta = \pm \sqrt{1 - \left( \dfrac{16}{49} \right)}\]
On further simplifying,
We get,
\[\Rightarrow \cos\ \theta = \pm \sqrt{\dfrac{49 – 16}{49}}\]
On subtracting,
We get,
\[\Rightarrow \cos\ \theta = \pm \sqrt{\dfrac{33}{49}}\]
Now on taking terms out of the radical sign,
We get,
\[\Rightarrow \cos\ \theta = \pm \dfrac{\sqrt{33}}{7}\]
Thus we get the value of \[\cos\ \theta = \pm \dfrac{\sqrt{33}}{7}\] .
The value of \[\cos\ \theta = \pm \dfrac{\sqrt{33}}{7}\] .

Note:These types of questions require grip over the concepts of trigonometry and identity . In this question , We are provided with a trigonometric expression in sine and cosine, then we need to use the formula and identity which contains both the given trigonometric function. While solving such trigonometric identities problems, we need to have a good knowledge about the trigonometric identities. One must know the correct trigonometric formulas and ratios to solve such problems .