Find the exact value of cos 75 degrees.
Isn't 75 degrees the sum of 45 and 30 degrees?
Isn't there a addition identity?
Cos(A+B)= ...
To find the exact value of cos 75 degrees, we can use the trigonometric identity for the cosine of a difference of angles:
cos(A - B) = cos A * cos B + sin A * sin B
In this case, we can rewrite 75 degrees as the sum of two angles: 45 degrees and 30 degrees.
75 degrees = 45 degrees + 30 degrees
Now, let's substitute these values into the cosine difference formula:
cos(75 degrees) = cos(45 degrees + 30 degrees)
Using the formula and the known values of cos 45 degrees (which is 1/√2) and cos 30 degrees (which is √3/2), we can calculate the exact value of cos 75 degrees.
cos(75 degrees) = cos(45 degrees) * cos(30 degrees) + sin(45 degrees) * sin(30 degrees)
cos(75 degrees) = (1/√2) * (√3/2) + (1/√2) * (1/2)
Multiply the terms:
cos(75 degrees) = √3/(2√2) + 1/(2√2)
To simplify this expression, we need to rationalize the denominators by multiplying both the numerator and denominator of each term by √2:
cos(75 degrees) = (√3 * √2) / (2√2 * √2) + (1 * √2) / (2√2 * √2)
cos(75 degrees) = (√6) / (4) + (√2) / (4)
Adding these terms together:
cos(75 degrees) = (√6 + √2) / 4
So, the exact value of cos 75 degrees is (√6 + √2) / 4.