Without using a calculator evaluate tan 105degrees
tan 105 = tan (60+45)
= (tan60 + tan45)/(1 - tan60 tan45)
= (√3 + 1)/(1 - √3)
6231*5.10
To evaluate tan 105 degrees without using a calculator, we can use the tangent addition formula.
The tangent addition formula states that tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a) * tan(b)).
In this case, we can use the formula to rewrite tan 105 degrees as the sum of two angles with known tangents.
First, we express 105 degrees as a sum of two angles that we can work with easily. We can choose 90 degrees and 15 degrees since the tangent of 90 degrees is undefined (infinity) and the tangent of 15 degrees is a known value.
So, tan 105 degrees = tan (90 degrees + 15 degrees).
Applying the tangent addition formula:
tan (90 degrees + 15 degrees) = (tan 90 degrees + tan 15 degrees) / (1 - tan 90 degrees * tan 15 degrees).
Notice that tan 90 degrees is undefined (infinity), so the above expression becomes:
tan (90 degrees + 15 degrees) = (infinity + tan 15 degrees) / (1 - infinity * tan 15 degrees).
Since multiplying infinity with any finite number gives infinity, the expression simplifies to:
tan (90 degrees + 15 degrees) = (infinity + tan 15 degrees) / infinity.
Dividing any finite number by infinity results in zero, so the expression further simplifies to:
tan (90 degrees + 15 degrees) = (0 + tan 15 degrees) / infinity.
Finally, since dividing a number by infinity results in zero, the answer is:
tan 105 degrees ≈ 0.
Therefore, without using a calculator, the approximate value of tan 105 degrees is 0.