Use half-angle formulas to find the exact value for the following
cot 75 degrees
75 degrees is one-half of 150 degrees, and you should know the sin and cosine of that. Let A/2=75 so that A = 150. Then use this half angle formula for cot A/2
cot(A/2) = +/- (1 + cos A)/sin A
You will have to pick the sign to go with the quadrant. In the first quadrant, it is positive
To find the exact value of cot 75 degrees using half-angle formulas, we can first reduce the angle to one of the commonly known angles such as 45 or 30 degrees. Since the half-angle of 60 degrees results in a multiple of 30 degrees, we can start with the half-angle formula for cot:
cot (A/2) = sqrt((1 + cos A) / (1 - cos A))
In this case, A is equal to 60 degrees (half of 120 degrees), so we need to find the value of cos(60). However, since 75 degrees is not a multiple of 30, we can split it into two commonly known angles:
75 degrees = 45 degrees + 30 degrees
Now, we can apply the sum formula for cot:
cot (A + B) = (cot A * cot B - 1) / (cot A + cot B)
Using A = 45 degrees and B = 30 degrees, we have:
cot 75 degrees = cot (45 degrees + 30 degrees)
cot 75 degrees = (cot 45 degrees * cot 30 degrees - 1) / (cot 45 degrees + cot 30 degrees)
Next, let's find the values of cot 45 degrees and cot 30 degrees:
cot 45 degrees = 1
cot 30 degrees = sqrt(3)
Now, substituting these values back into the equation:
cot 75 degrees = (1 * sqrt(3) - 1) / (1 + sqrt(3))
Simplifying further:
cot 75 degrees = (sqrt(3) - 1) / (1 + sqrt(3))
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
cot 75 degrees = ((sqrt(3) - 1) / (1 + sqrt(3))) * ((1 - sqrt(3)) / (1 - sqrt(3)))
Expanding:
cot 75 degrees = (sqrt(3) - sqrt(3) - 1 + sqrt(3)) / (1 - 3)
Further simplifying:
cot 75 degrees = (2sqrt(3) - 1) / (-2)
Finally, we can simplify the expression by dividing both the numerator and denominator by -2:
cot 75 degrees = (1 - 2sqrt(3)) / 2
Therefore, the exact value of cot 75 degrees using half-angle formulas is (1 - 2sqrt(3)) / 2.