Question

Find the exact value by using ahalf-angle formula.[?] --cos 75°

Answer 1

Answer::

`\cos 75\degree=\frac{\sqrt[]{2-\sqrt[]{3}}}{2}`

Explanation:

By the half-angle formula:

`\cos \mleft((\theta)/(2)\mright)=\pm\sqrt[]{(1+\cos\theta)/(2)}`

Let θ=150°, therefore:

`\begin{gathered} \cos ((150\degree)/(2))=\sqrt[]{(1+\cos150\degree)/(2)} \\ \cos (150)\degree=-\cos (180\degree-150\degree)=-\cos 30\degree \\ \implies\cos ((150\degree)/(2))=\sqrt[]{(1+\cos150\degree)/(2)}=\sqrt[]{(1-\cos30\degree)/(2)} \end{gathered}`

Now, cos 30 = √3/2, thus:

`\begin{gathered} =\sqrt[]{\frac{1-\frac{\sqrt[]{3}}{2}}{2}} \\ \text{Multiply both the denominator and numerator by 2} \\ =\sqrt[]{\frac{2-\sqrt[]{3}}{4}} \\ =\frac{\sqrt{2-\sqrt[]{3}}}{√(4)} \end{gathered}`

The exact value of cos 75° is:

`\cos 75\degree=\frac{\sqrt[]{2-\sqrt[]{3}}}{2}`