Find the exact value of tan (11pi/12)
recall cos (2A) = 2cos^2 A - 1 or 1 - 2sin^2 A
cos 30 = 2cos^2 15 - 1
√3/2 + 1 = 2cos^2 15
(√3 + 2)/4 = cos^2 15
cos 15° = √(√3 + 2)/2
cos 30 = 1 - 2sin^2 15
2sin^2 15 = 1 - √3/2 = (2 - √3)/2
sin^2 15 = (2 - √3)/4
sin 15 = √(2 - √3)/2
tan 15 = sin 15/cos 15 = √(2 - √3)/2 / √(2 + √3)/2
= √(2 - √3) / √(2 + √3)
so tan 165 = - tan 15° = - √(2 - √3) / √(2 + √3)
seems messier.
tan(x/2) = (1-cosx)/sinx = (1 - √3/2) / (-1/2) = (√3-2)
Note that
(1-√3)/(1+√3) = (1-√3)^2 / (1-3) = (1+3-2√3)/-2 = √3-2
-√(2-√3)/(2+√3) = -√((2-√3)^2)/(4-3)) = -(2-√3) = √3-2
11π/12 is the same as 165°
tan 11π/12 = tan 165°
= - tan 15°
Now tan15° = tan(45-30)°
= (tan45 - tan30) / (1 + tan45tan30)
= (1 - 1/√3) / (1 + (1)(1/√3) )
multiply top and bottom by √3
= (√3 - 1)/(√3 + 1)
so tan 11π/12 = -tan15 = (1 - √3)/(√3 + 1) , I changed the sign on the top
can you show me by using the half-angle formula
To find the exact value of tan (11π/12), we need to follow a few steps.
Step 1: Determine the reference angle.
The reference angle is the positive acute angle formed between the terminal side of the angle and the x-axis. In this case, the angle is 11π/12, which is greater than π/2 (90 degrees) but less than π (180 degrees). The reference angle can be found by subtracting π from the original angle:
Reference Angle = 11π/12 - π = (11π - 12π)/12 = -π/12
Step 2: Determine the sign of the tangent.
Since the angle 11π/12 falls in Quadrant II (where cosine is negative, and sine is positive), the tangent will be negative.
Step 3: Find the tangent value of the reference angle.
To find the tangent of the reference angle, we use the trigonometric identity:
tan(θ) = sin(θ) / cos(θ)
Since the reference angle is -π/12, we can find the sine and cosine values using the unit circle or trigonometric ratios.
For the reference angle -π/12, we can use the exact values from the unit circle:
sin(-π/12) = -1/2
cos(-π/12) = √3 / 2
Step 4: Calculate the tangent value.
Now, we can find the tangent value by dividing the sine value by the cosine value:
tan(-π/12) = (sin(-π/12)) / (cos(-π/12)) = (-1/2) / (√3 / 2)
To get rid of the fractions in the denominator, we can multiply the numerator and denominator by the conjugate of the denominator (√3 / 2):
tan(-π/12) = (-1/2) * (2 / √3) = -1 / √3
To rationalize the denominator, we multiply the numerator and denominator by √3:
tan(-π/12) = (-1 / √3) * (√3 / √3) = -√3 / 3
So, the exact value of tan (11π/12) is -√3 / 3.