Use the sum or difference indentity to find the exact value of tan 105 degrees.
Since 60+45=105,
would you like to give it a try and show me what you've got.
no
To find the exact value of tan 105 degrees using the sum or difference identity, we need to express 105 degrees as the sum or difference of angles whose tangent values we already know. The sum or difference identity for tangent is as follows:
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A * tan B)
Let's express 105 degrees as the sum or difference of angles we know the tangent values for. Since tan 45 degrees = 1 and tan 60 degrees = √3, we can express 105 degrees in terms of these angles.
105 degrees = 45 degrees + 60 degrees
Now we can substitute these values into the sum identity:
tan(105 degrees) = tan(45 degrees + 60 degrees)
Using the sum identity:
tan(45 degrees + 60 degrees) = (tan 45 degrees + tan 60 degrees) / (1 - tan 45 degrees * tan 60 degrees)
Substituting the known values:
tan(105 degrees) = (1 + √3) / (1 - 1 * √3)
Simplifying the expression:
tan(105 degrees) = (1 + √3) / (1 - √3)
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
tan(105 degrees) = ((1 + √3) / (1 - √3)) * ((1 + √3) / (1 + √3))
Expanding the numerator and denominator:
tan(105 degrees) = (1 + 2√3 + 3) / (1 - 3)
Simplifying the expression:
tan(105 degrees) = (4 + 2√3) / -2
Finally, we get the exact value of tan 105 degrees as:
tan(105 degrees) = -(2 + √3)