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valentinak56 [21]
3 months ago
11

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

Mathematics
1 answer:
Brrunno [24]3 months ago
6 0

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}

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