Question

Find the value of $\cos 18$ degrees?

Answer 1

Hint: As we can see that the above question is related to trigonometry. As sine and cosine are trigonometric ratios. We will apply the trigonometric identities to solve the above question. We know that ${\sin }^2\theta +{\cos }^2\theta =1$ . With the help of this identity we will find the value of the above question.

Complete step-by-step answer:
Let us assume that $\theta =18$ .
We will apply the above identity and we can write it as ${\sin }^{2}18+{\cos }^{2}18=1$ .
We have to find the value of cosine, so we will isolate the term,
 ${\cos }^{2}18=1-{\sin }^{2}18$ .
The value of ${\sin }^{2}18={\left({\displaystyle \frac{-1+\sqrt{5}}{4}}\right)}^2$ , so by putting this in the formula we can write
 ${\cos }^{2}18=1-{\left({\displaystyle \frac{-1+\sqrt{5}}{4}}\right)}^2$ .
Now we will solve the above i.e.
 ${\cos }^{2}18=1-\left({\displaystyle \frac{1^2+{\sqrt{5}}^2-2\sqrt{5}}{16}}\right)=1-\left({\displaystyle \frac{6-2\sqrt{5}}{16}}\right)$ .
By breaking the bracket we have
 ${\displaystyle \frac{16-6+2\sqrt{5}}{16}}={\displaystyle \frac{10+2\sqrt{5}}{16}}$ .
We can write $\cos 18={\displaystyle \frac{\sqrt{10+2\sqrt{5}}}{4}}$ .
Hence the required value is $\cos 18={\displaystyle \frac{\sqrt{10+2\sqrt{5}}}{4}}$ .
So, the correct answer is “ $\cos 18={\displaystyle \frac{\sqrt{10+2\sqrt{5}}}{4}}$ .”.

Note: Before solving this kind of question we should be fully aware of the trigonometric identities and the formulas. We can find the value of $\sin 18$ , by assuming $\theta =18$ .. We can write this as $2\theta +3\theta =90(5\theta =5\times 18=90)$ . Then we can write it as $2A=90-3A$ .
We will take sin on both sides and then apply the identity $\sin 2A=2\sin A\cos A=4{\cos }^{3}A-3\cos A$ . By substituting the values and applying the formula we can get the value. We can also find the value in radian as it is the same in radian and degrees.