Find the exact value of tan 5pi/12 using the half-angle identity.
To find the exact value of tan(5π/12) using the half-angle identity, we need to recall the half-angle formula for tangent:
tan(x/2) = ±√((1 - cosx) / (1 + cosx))
In this case, x = 5π/6. Let's start by finding cos(5π/6):
Using the angle sum formula, we can write cos(5π/6) as cos(2π/3 + π/6):
cos(2π/3 + π/6) = cos(2π/3)cos(π/6) - sin(2π/3)sin(π/6)
Recall that cos(2π/3) = -1/2 and sin(2π/3) = √3/2:
cos(5π/6) = (-1/2)(√3/2) - (√3/2)(1/2)
= -√3/4 - √3/4
= -√3/2
Next, substitute cos(5π/6) into the half-angle formula for tangent:
tan(5π/12) = ±√((1 - cos(5π/6)) / (1 + cos(5π/6)))
Doing the calculation, we get:
tan(5π/12) = ±√((1 + √3/2) / (1 - √3/2))
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
tan(5π/12) = ±√((1 + √3/2) * (1 + √3/2) / (1 - √3/2) * (1 + √3/2))
Simplifying further, we get:
tan(5π/12) = ±√((1 + 2√3/2 + 3/4) / (1 - (3/2)))
= ±√((7/4 + 2√3/2) / (-1/2))
= ±√((-7/4 - 2√3/2) / 1)
= ±√(-7/4 - √3/2)
Therefore, the exact value of tan(5π/12) using the half-angle identity is ±√(-7/4 - √3/2).
To find the exact value of tan(5π/12) using the half-angle identity, we will start by using the double-angle identity for tangent.
The half-angle identity for tangent states that:
tan(2θ) = (2tan(θ))/(1 - tan²(θ))
Let's set θ = 5π/6 and divide it into two equal parts:
θ = (5π/6)/2 = 5π/12
Now, using the half-angle identity, we have:
tan(5π/6) = (2tan(5π/12))/(1 - tan²(5π/12))
Our goal is to find tan(5π/12), so we can rearrange the equation:
2tan(5π/12) = tan(5π/6) - tan²(5π/12) * tan(5π/6)
tan(5π/12) * (2 + tan²(5π/6)) = tan(5π/6)
Now, we need to find the values of tan(5π/6) and tan(5π/12):
Here are the exact values of tan(5π/6) and tan(5π/12):
tan(5π/6) = √3
Now, we can substitute these values back into our equation to solve for tan(5π/12):
tan(5π/12) * (2 + tan²(5π/6)) = √3
Now, rearrange the equation to solve for tan(5π/12):
tan²(5π/12) = (√3 - 2) / (√3)
To find the value of tan(5π/12), you can take the square root of both sides:
tan(5π/12) = ±√[(√3 - 2) / (√3)]
So, the exact value of tan(5π/12) is ±√[(√3 - 2) / (√3)].
tan 5pi/6 = -1/√3
tan x/2 = (1-cos(x))/sin(x)
cos 5pi/6 = -√3/2
sin 5pi/6 = 1/2
tan 5pi/12 = (1 + √3/2)/(1/2)
= 2 + √3