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

Find the exact value by using a half-angle formula. cos 75° = 2-√3-example-1
User Null Set
by
2.8k points

1 Answer

22 votes
22 votes

Answer:


\frac{\sqrt{2-√(3)}}{2}

Explanation:

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

So plugging in the values we get:


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

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


cos(75) = \sqrt{(1-(√(3))/(2))/(2)\\

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


cos(75) = \sqrt{((2)/(2)-(√(3))/(2))/(2)\\

Combine the numerator into one fraction


cos(75) = \sqrt{((2-√(3))/(2))/(2)\\

Keep, change, flip


cos(75) = \sqrt{(2-√(3))/(2)*(1)/(2)}

Multiply:

cos(75) = \sqrt{(2-√(3))/(4)

We can distribute the square root across the division:


\frac{\sqrt{2-√(3)}}{√(4)}

Simplify the denominator


\frac{\sqrt{2-√(3)}}{2}

User Molerat
by
2.6k points