Question

Find the exact value by using a half-angle formula. cos 75° = 2-√3

Answer 1

Answer:

$\frac{\sqrt{2-\sqrt{3}}}{2}$

Step-by-step explanation:

since 75 is half of 150, we can use the half angle formula of cos defined as: $cos(\frac{\theta}{2}) = \sqrt{\frac{1+cos(\theta)}{2}}$ where theta=150

So plugging in the values we get:

$cos(75) = \sqrt{\frac{1+cos(150)}{2}}$

Now using the unit circle, we can solve for cos(150)

$cos(75) = \sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}$

Now multiply the 1 by 2/2 to combine the numerator into one fraction

$cos(75) = \sqrt{\frac{\frac{2}{2}-\frac{\sqrt{3}}{2}}{2}$

Combine the numerator into one fraction

$cos(75) = \sqrt{\frac{\frac{2-\sqrt{3}}{2}}{2}$

Keep, change, flip

$cos(75) = \sqrt{\frac{2-\sqrt{3}}{2}*\frac{1}{2}}$

Multiply:
$cos(75) = \sqrt{\frac{2-\sqrt{3}}{4}$

We can distribute the square root across the division:

$\frac{\sqrt{2-\sqrt{3}}}{\sqrt{4}}$

Simplify the denominator

$\frac{\sqrt{2-\sqrt{3}}}{2}$