find the exact value of:
tan 7pie/12
drwls that's wrong :O
tan 7pie/12 = tan (3pie/12 + 4pie/12) *break it down*
= tan (pie/4 + pie/3)
*find tan pie/4 and tan pie/3 on unit circle*
= 1 + square root of 3 *answer*
thats the exact value =]
-Cyborg03
To find the exact value of tan (7π/12), you can use the tangent half-angle formula along with some trigonometric identities.
The tangent half-angle formula states that tan (θ/2) = ± √((1 - cos θ) / (1 + cos θ)).
First, let's find cos (7π/6):
7π/6 = (3π/2) + (π/6)
cos (7π/6) = cos ((3π/2) + (π/6))
Using the sum angle formula, cos (a + b) = cos a * cos b - sin a * sin b, we have:
cos (7π/6) = cos (3π/2) * cos (π/6) - sin (3π/2) * sin (π/6)
= (0) * (√3/2) - (-1) * (1/2)
= 0 + 1/2
= 1/2
Now, substitute this value into the tangent half-angle formula:
tan (7π/12) = ± √((1 - cos (7π/6)) / (1 + cos (7π/6)))
= ± √((1 - 1/2) / (1 + 1/2))
= ± √(1/2 / 3/2)
= ± √(1/3)
= ± 1/√3
To rationalize the denominator, multiply both the numerator and the denominator by √3:
= ± (1/√3) * (√3/√3)
= ± (√3 / 3)
Therefore, the exact value of tan (7π/12) is ± (√3 / 3).
To find the exact value of tan(7π/12), we can use the half-angle formula for tangent. The half-angle formula states that tan(θ/2) = (1 - cosθ) / sinθ.
1. First, let's find the values of cos(7π/6) and sin(7π/6).
To find cos(7π/6), we need to determine the reference angle. The reference angle is calculated by subtracting π from the given angle, resulting in (7π/6) - π = π/6.
We know that cos(π/6) = √3/2 since it is a commonly known value for the 30-degree reference angle.
Since the cosine function is negative in the second quadrant, cos(7π/6) = -√3/2.
To find sin(7π/6) using the Pythagorean identity, we square sin(π/6) and subtract it from 1, resulting in:
sin(7π/6) = -sin(-π/6) = -sin(π/6) = -1/2.
2. Substitute the values of cos(7π/6) and sin(7π/6) into the half-angle formula:
tan(7π/12) = (1 - cos(7π/6)) / sin(7π/6)
= (1 - (-√3/2)) / (-1/2)
= (1 + √3/2) / (-1/2)
= (1 + √3/2) × (-2/1)
= -2(1 + √3) / 2
= -1 - √3.
Therefore, the exact value of tan(7π/12) is -1 - √3.
pi/12 is 15 degrees
7 pi/12 is 105 degrees
tan (7 pi/12) = -1/tan (pi/12)
The tangent is -2 -sqrt 3 = -3.73205...