Use the sum or difference identities to find the exact value of each trigonometric function.
sec pi/12
Thanks :)
To find the exact value of sec(pi/12) using the sum or difference identities, we first need to identify a sum or difference identity that involves the cosine function. The secant function, sec(x), is the reciprocal of the cosine function, so we'll use the identity:
sec(x) = 1/cos(x)
Now, let's express pi/12 as a sum or difference of angles that we can work with. We'll use the fact that pi/4 is equivalent to 3pi/12.
pi/12 = pi/4 - pi/6
Now, we can use the sum or difference identity for cosine:
cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
Let's assign a = pi/4 and b = pi/6, and plug them into the formula:
cos(pi/12) = cos(pi/4 - pi/6) = cos(pi/4)cos(pi/6) + sin(pi/4)sin(pi/6)
Now, substituting the values of cos(pi/4) = 1/sqrt(2) and sin(pi/4) = 1/sqrt(2), and cos(pi/6) = sqrt(3)/2 and sin(pi/6) = 1/2, we get:
cos(pi/12) = (1/sqrt(2))(sqrt(3)/2) + (1/sqrt(2))(1/2)
Simplifying this expression:
cos(pi/12) = sqrt(3)/(2*sqrt(2)) + 1/(2*sqrt(2))
Now, we need to rationalize the denominator by multiplying throughout by sqrt(2):
cos(pi/12) = (sqrt(3)/(2*sqrt(2))) * (sqrt(2)/sqrt(2)) + (1/(2*sqrt(2))) * (sqrt(2)/sqrt(2))
cos(pi/12) = sqrt(6)/4 + sqrt(2)/4
Finally, taking the reciprocal of cos(pi/12) to find sec(pi/12):
sec(pi/12) = 1/cos(pi/12) = 1/(sqrt(6)/4 + sqrt(2)/4)
To simplify this further, we can multiply the numerator and denominator by the conjugate of the denominator, which is sqrt(6)/4 - sqrt(2)/4:
sec(pi/12) = (1/(sqrt(6)/4 + sqrt(2)/4)) * (sqrt(6)/4 - sqrt(2)/4)
sec(pi/12) = (sqrt(6)/4 - sqrt(2)/4) / ((sqrt(6)/4)^2 - (sqrt(2)/4)^2)
Expanding the denominator, we have:
sec(pi/12) = (sqrt(6)/4 - sqrt(2)/4) / (6/16 - 2/16)
sec(pi/12) = (sqrt(6)/4 - sqrt(2)/4) / (4/16)
Dividing the numerator by 4/16 gives us:
sec(pi/12) = (sqrt(6)/4 - sqrt(2)/4) * (16/4)
sec(pi/12) = 4sqrt(6)/4 - 4sqrt(2)/4
simplifying:
sec(pi/12) = sqrt(6) - sqrt(2)
Therefore, the exact value of sec(pi/12) is sqrt(6) - sqrt(2).