Use an Addition or Subtraction Formula to find the exact value of the expression:
tan7π/12
7/12 = 1/3 + 1/4
now review your addition formula for tan(x+y)
To find the exact value of the expression tan(7π/12), we can use the tangent addition formula:
tan(A + B) = (tanA + tanB) / (1 - tanA * tanB)
First, let's rewrite the angle 7π/12 as the sum of two known angles. We know that π/6 + π/4 = π/12, so we can rewrite 7π/12 as:
7π/12 = π/6 + π/4
Now, let's substitute these values into the tangent addition formula:
tan(7π/12) = tan(π/6 + π/4) = (tan(π/6) + tan(π/4)) / (1 - tan(π/6) * tan(π/4))
Next, let's find the tangent values of π/6 and π/4.
For π/6, we know that sin(π/6) = 1/2 and cos(π/6) = sqrt(3)/2. Therefore, tan(π/6) = sin(π/6) / cos(π/6) = (1/2) / (sqrt(3)/2) = 1 / sqrt(3) = sqrt(3)/3.
For π/4, we know that sin(π/4) = cos(π/4) = sqrt(2)/2. Therefore, tan(π/4) = sin(π/4) / cos(π/4) = (sqrt(2)/2) / (sqrt(2)/2) = 1.
Now, let's substitute these values back into the formula:
tan(7π/12) = (sqrt(3)/3 + 1) / (1 - (sqrt(3)/3)(1))
Simplifying further:
tan(7π/12) = (sqrt(3)/3 + 1) / (1 - sqrt(3)/3)
To simplify the expression, we can multiply the numerator and denominator by (3 + sqrt(3)):
tan(7π/12) = ((sqrt(3)/3 + 1)(3 + sqrt(3))) / ((1 - sqrt(3)/3)(3 + sqrt(3)))
Expanding and simplifying:
tan(7π/12) = (sqrt(3) + 3 + sqrt(3) + 3sqrt(3)) / (3 - sqrt(3) + 3 - sqrt(3)/3)
Continuing to simplify:
tan(7π/12) = (4sqrt(3) + 6) / (6 - sqrt(3))
Finally, we can rationalize the denominator by multiplying both numerator and denominator by the conjugate of the denominator (6 + sqrt(3)):
tan(7π/12) = ((4sqrt(3) + 6)(6 + sqrt(3))) / ((6 - sqrt(3))(6 + sqrt(3)))
Expanding and simplifying:
tan(7π/12) = (24sqrt(3) + 42 + 4sqrt(3) + 6sqrt(3)) / (36 - 3)
Further simplification gives the exact value of the expression:
tan(7π/12) = (30sqrt(3) + 42) / 33