<\/span><\/h2>\n<\/p>\n
To find the exact value of tan \u03c0\/12, we need to employ trigonometric identities and special angle values. The angle \u03c0\/12 is not considered a special angle, so its exact value cannot be determined with ease. However, by using certain trigonometric identities, we can express tan \u03c0\/12 in terms of other known angles and trigonometric values.<\/p>\n
Let’s use the sum angle identity for the tangent function, which states that tan(A + B) = (tan A + tan B) \/ (1 – tan A tan B). By setting A = \u03c0\/6 (30 degrees) and B = \u03c0\/12 (15 degrees), we can rearrange the formula as:<\/p>\n
tan \u03c0\/12 = (tan \u03c0\/6 + tan \u03c0\/12) \/ (1 – tan \u03c0\/6 tan \u03c0\/12)<\/p>\n
But what are the exact values of tan \u03c0\/6 and tan \u03c0\/12? Luckily, the tangent of \u03c0\/6 is a special angle value, since it corresponds to a 30-degree angle, and its exact value is 1\/\u221a3. By substituting this value into the formula, we can simplify it further.<\/p>\n
tan \u03c0\/12 = (1\/\u221a3 + tan \u03c0\/12) \/ (1 – (1\/\u221a3)(tan \u03c0\/12))<\/p>\n
Multiplying the numerator and denominator by \u221a3 to eliminate the square root in the denominator, we get:<\/p>\n
\u221a3 tan \u03c0\/12 = (\u221a3 + \u221a3 tan \u03c0\/12) \/ (\u221a3 – tan \u03c0\/12)<\/p>\n
Next, we can cross-multiply and rearrange the terms:<\/p>\n
\u221a3 tan \u03c0\/12 – \u221a3 tan \u03c0\/12 tan \u03c0\/12 = \u221a3 + \u221a3 tan \u03c0\/12<\/p>\n
Factoring out \u221a3 tan \u03c0\/12, we obtain:<\/p>\n
(\u221a3 – tan \u03c0\/12) tan \u03c0\/12 = \u221a3<\/p>\n
Finally, solving for tan \u03c0\/12:<\/p>\n
tan \u03c0\/12 = \u221a3 \/ (\u221a3 – tan \u03c0\/12)<\/b><\/p>\n
This is the exact value of the tangent of the angle \u03c0\/12. It may not appear in a simpler form, but it accurately represents the ratio of the lengths of the opposite and adjacent sides in a right triangle with a \u03c0\/12 angle.<\/p>\n