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Calculus/pre cal 12th Circular functions II


Find the exact value of cos 75°

is is known that cos²(75°) = (1/2)(1 + cos(150²)).

Since 150° is 30° from 180°, it is known that cos(150°)=-cos(30°).
Since it is known on a 30-60-90 triangle the sides have length 1 and root(3)
with a hypotenuse of 2, the cos(30°) = root(3)/2.  Thus, -cos(30°) = -root(3)/2.
Going back another step, cos(150°) = -cos(30°), so cos(150°) = -root(3)/2.

Putting this into the formula gives cos²(75°) = (1/2)(1 + -root(3)/2).
This can be rewritten as cos²(75°) = ((2 + -root(3))/4

Taking the squareroot of both sides gives cos(75°) = root(2 - root(3))/2.
Since 75° is in the 1st quadrant, the cos() of that angle is positive.
Since root(3)<2, that is a positive number.

Checking this out in Excel, both cos(75*pi()/180) and =(2-3^0.5)^0.5 /2  are 0.2588.


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