use a half angle formula to find the exact value of the expression.
tan112.5 degrees
The half angle formula for tangent is given by:
tan(x/2) = (1 - cos x) / sin x
Let's use this formula to find the exact value of tan(112.5 degrees):
First, we need to express 112.5 degrees as a sum of two angles whose tangents we know.
112.5 = 45 + 67.5
Now, we can rewrite tan(112.5 degrees) in terms of these angles:
tan(112.5) = tan(45 + 67.5) = (tan(45) + tan(67.5)) / (1 - tan(45)tan(67.5))
We know that tan(45 degrees) = 1 and tan(67.5 degrees) = tan(135/2) from the identities.
Plugging these values in, we get:
tan(112.5) = (1 + tan(135/2)) / (1 - 1 * tan(135/2))
By the half angle formula, we know that:
tan(135/2) = (1 - cos 135) / sin 135 = (1 - (-√2/2)) / (-√2/2) = (1 + √2/2) / (-√2/2) = - (1 + √2) / √2
Now, substituting back into the equation:
tan(112.5) = (1 - cos 51 ') / sin 51' = (1 + (-1 - √2) / √2) / (-√2/2) = (-√2) / (-√2/2) = 2
Therefore, the exact value of tan(112.5 degrees) is 2.