Answer:
[tex]\frac{\sqrt{2-\sqrt{3}}}{2}[/tex]
Step-by-step explanation:
since 75 is half of 150, we can use the half angle formula of cos defined as: [tex]cos(\frac{\theta}{2}) = \sqrt{\frac{1+cos(\theta)}{2}}\\[/tex] where theta=150
So plugging in the values we get:
[tex]cos(75) = \sqrt{\frac{1+cos(150)}{2}}[/tex]
Now using the unit circle, we can solve for cos(150)
[tex]cos(75) = \sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}\\[/tex]
Now multiply the 1 by 2/2 to combine the numerator into one fraction
[tex]cos(75) = \sqrt{\frac{\frac{2}{2}-\frac{\sqrt{3}}{2}}{2}\\[/tex]
Combine the numerator into one fraction
[tex]cos(75) = \sqrt{\frac{\frac{2-\sqrt{3}}{2}}{2}\\[/tex]
Keep, change, flip
[tex]cos(75) = \sqrt{\frac{2-\sqrt{3}}{2}*\frac{1}{2}}[/tex]
Multiply:
[tex]cos(75) = \sqrt{\frac{2-\sqrt{3}}{4}[/tex]
We can distribute the square root across the division:
[tex]\frac{\sqrt{2-\sqrt{3}}}{\sqrt{4}}[/tex]
Simplify the denominator
[tex]\frac{\sqrt{2-\sqrt{3}}}{2}[/tex]
