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

If sin⁡ θ=47\sin\ \theta = \dfrac{4}{7}.what is cos⁡ θ\cos\ \theta ?

Explanation

Solution

In this question, given that sin⁡ θ=47\sin\ \theta = \dfrac{4}{7} then, we need to find the value of cos⁡ θ\cos\ \theta . By using trigonometric identities , we can find the cos⁡ θ\cos\ \theta . First we can use Pythagoras trigonometric identity  cos⁡2θ+sin⁡2θ=1\ \cos^{2}\theta + \sin^{2}\theta = 1. Then we need to subtract sin⁡2θ\sin^{2}\theta on both sides. Then on taking square root and simplifying, we get the expression in the form of cos⁡ θ\cos\ \theta and sin⁡ θ\sin\ \theta. Then we need to substitute the value of sin⁡ θ\sin\ \theta and on further simplifying, we can find the value of cos⁡ θ\cos\ \theta .

Complete step by step answer:
Given, sin⁡ θ=47\sin\ \theta = \dfrac{4}{7}
Here we need to find cos⁡ θ\cos\ \theta .
By using Pythagoras trigonometric identity,
⇒cos⁡2θ+sin⁡2θ=1\Rightarrow \cos^{2}\theta + \sin^{2}\theta = 1
On subtracting sin⁡2θ\sin^{2}\theta on both sides,
We get,
⇒cos⁡2θ+sin⁡2θ−sin⁡2θ=1 −sin⁡2θ\Rightarrow \cos^{2}\theta + \sin^{2}\theta - \sin^{2}\theta = 1\ - \sin^{2}\theta
On simplifying ,
We get,
⇒cos⁡2θ=1−sin⁡2θ\Rightarrow \cos^{2}\theta = 1 - \sin^{2}\theta
On taking square root on both sides,
⇒ cos⁡ θ=±1−sin⁡2θ\Rightarrow \ \cos\ \theta = \pm \sqrt{1 - \sin^{2}\theta}
Now on substituting the value of sin⁡ θ\sin\ \theta ,
We get,
⇒cos⁡ θ=±1−(47)2\Rightarrow \cos\ \theta = \pm \sqrt{1 - \left( \dfrac{4}{7} \right)^{2}}
On simplifying,
We get,
⇒cos⁡ θ=±1−(1649)\Rightarrow \cos\ \theta = \pm \sqrt{1 - \left( \dfrac{16}{49} \right)}
On further simplifying,
We get,
⇒cos⁡ θ=±49–1649\Rightarrow \cos\ \theta = \pm \sqrt{\dfrac{49 – 16}{49}}
On subtracting,
We get,
⇒cos⁡ θ=±3349\Rightarrow \cos\ \theta = \pm \sqrt{\dfrac{33}{49}}
Now on taking terms out of the radical sign,
We get,
⇒cos⁡ θ=±337\Rightarrow \cos\ \theta = \pm \dfrac{\sqrt{33}}{7}
Thus we get the value of cos⁡ θ=±337\cos\ \theta = \pm \dfrac{\sqrt{33}}{7} .
The value of cos⁡ θ=±337\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 .