use a half-angle identity to find the exact value of tan 105 degrees.
To find the exact value of tan 105 degrees using a half-angle identity, we can start by using the half-angle identity for the tangent function, which is:
tan(θ/2) = ±√((1 - cosθ) / (1 + cosθ))
In this case, we want to find the value of tan 105 degrees, which is larger than 90 degrees. To make it easier, we can rewrite 105 degrees as the sum of 90 degrees and 15 degrees:
105 degrees = 90 degrees + 15 degrees
Now, let's substitute this value into the half-angle identity:
tan(105/2) = ±√((1 - cos(90 + 15)) / (1 + cos(90 + 15)))
Since 90 + 15 = 105, the expression becomes:
tan(105/2) = ±√((1 - cos 105) / (1 + cos 105))
At this point, we need to find the cosine of 105 degrees. To do so, we can use the fact that cosine is an even function, meaning cos(-x) = cos(x). Therefore, we have:
cos(105) = cos(-105)
And since cos(180 - x) = -cos(x), we can rewrite it as:
cos(105) = -cos(75)
Now we can substitute this value back into the half-angle identity:
tan(105/2) = ±√((1 - (-cos 75)) / (1 + (-cos 75)))
Simplifying:
tan(105/2) = ±√((1 + cos 75) / (1 - cos 75))
Now, to find the exact value, we need to know the value of cos 75 degrees. To do this, we can use another trigonometric identity, such as the half-angle identity for cosine:
cos(θ/2) = ±√((1 + cosθ) / 2)
Applying this to 75 degrees:
cos(75/2) = ±√((1 + cos 150) / 2)
Since cos 150 = -cos 30 and cos 30 = √3 / 2, we have:
cos(75/2) = ±√((1 - (−√3 / 2)) / 2)
Simplifying:
cos(75/2) = ±√((1 + √3) / 2)
Finally, substituting this value into the expression for tan(105/2):
tan(105/2) = ±√((1 + √3) / (1 - √((1 + √3) / 2)))
This is the exact value of tan 105 degrees, using the half-angle identity.
To find the exact value of tan 105 degrees using a half-angle identity, we can use the formula for tangent of half an angle:
tan(θ/2) = ±√((1-cosθ) / (1+cosθ))
In this case, we want to find the value of tan 105 degrees, which is equivalent to finding the value of tan (210/2).
Step 1: Find the value of cos 210 degrees
Using the unit circle or trigonometric identities, we know that cos 210 degrees = cos (180 + 30) degrees = -cos 30 degrees = -√3/2
Step 2: Substitute the value of cos 210 degrees into the formula for tangent of half an angle
tan(105 degrees) = tan(210/2) = ±√((1-cos 210 degrees) / (1+cos 210 degrees))
tan(105 degrees) = ±√(1- (-√3/2)) / (1 + (-√3/2))
Step 3: Simplify the expression further
Since tan is positive in the second quadrant (where 105 degrees is located), we take the positive root:
tan(105 degrees) = √(1 + √3/2) / (1 - √3/2)
Step 4: Rationalize the denominator
To rationalize the denominator (i.e., remove the square root from the denominator), we multiply both the numerator and denominator by the conjugate of the denominator:
tan(105 degrees) = (√(1 + √3/2) / (1 - √3/2)) * ((1 + √3/2) / (1 + √3/2))
tan (105 degrees) = (√(1 + √3/2) * (1 + √3/2)) / (1 - (√3/2)^2)
tan (105 degrees) = (√(1 + √3/2) * (1 + √3/2)) / (1 - 3/4)
tan (105 degrees) = (√(1 + √3/2) * (1 + √3/2)) / (1/4)
tan (105 degrees) = 4 * (√(1 + √3/2) * (1 + √3/2))
Hence, the exact value of tan 105 degrees using a half-angle identity is 4 * (√(1 + √3/2) * (1 + √3/2)).
we know that
tan 2A = 2tanA/(1 - tan^2 A)
so tan 210 = 2tan105/(1 - tan^2 105)
you should know that
tan 210 = tan 30° = 1/√3
let tan 105 = x
then
1/√3 = 2x/(1-x^2)
2√3x = 1 - x^2
x^2 + 2√3 - 1 = 0
x = (-2√3 ± √16)/2
= -√3 ± 2
but obviously tan105 is negative, ( 105° is in II )
so tan 105° = -√3-2