Calculating the Exact Value of tan(5π/12): Methods and Insights
In trigonometry, determining the exact value of trigonometric functions such as tangent is a fundamental task. This article explores the exact value of tan(5π/12), employing several methods including angle sum identities, half-angle formulas, and tangent identities. We will present the results and derive step-by-step solutions to ensure a deep understanding of the subject matter.
Method 1: Using Angle Sum Identity
To find the exact value of tan(5π/12), we can use the angle sum identity for tangent. We start by expressing 5π/12 as the sum of two angles whose tangents are known:
5π/12 π/3 π/4
Using the tangent sum identity:
tan(a b) (tan a tan b) / (1 - tan a tan b)
Let a π/3 and b π/4. We know:
tan(π/3) √3
tan(π/4) 1
Substituting these values into the identity:
tan(5π/12) (tan(π/3) tan(π/4)) / (1 - tan(π/3) * tan(π/4))
(sqrt{3} 1) / (1 - sqrt{3} * 1)
(sqrt{3} 1) / (1 - sqrt{3})
To simplify this expression, we multiply the numerator and denominator by the conjugate of the denominator:
(sqrt{3} 1)(1 sqrt{3}) / (1 - sqrt{3})(1 sqrt{3})
(sqrt{3} 1)(1 sqrt{3}) / (1 - 3)
(sqrt{3} 1)(1 sqrt{3}) / (-2)
Expanding the numerator:
(3 2sqrt{3} 1) / (-2)
(4 2sqrt{3}) / (-2)
-2 - sqrt{3}
Method 2: Using Half-Angle Formula
Another approach to finding tan(5π/12) is using the half-angle formula for tangent:
tan θ/2 ±√(1 - cos θ) / (1 cos θ)
Since 5π/12 75° 45° 30°, we can use the tangent sum identity again:
tan 5π/12 tan(75°) tan(45° 30°) (tan 45° tan 30°) / (1 - tan 45° tan 30°)
(1 1/√3) / (1 - 1/√3) (sqrt{3} 1) / (√3 - 1)
Multiplying numerator and denominator by the conjugate of the denominator:
(sqrt{3} 1)(√3 1) / (sqrt{3} - 1)(sqrt{3} 1) (3 2sqrt{3} 1) / (3 - 1) (4 2sqrt{3}) / 2 2 sqrt{3}
Method 3: Using Cotangent and Angle Subtraction
A third method involves using the cotangent and angle subtraction techniques:
tan(5π/12) tan(60° - 15°) cot(15°) 1 / tan(15°)
tan(15°) (1 - cos30°) / sin30° (1 - √3/2) / (1/2) 2 - √3
Therefore:
tan(5π/12) 1 / (2 - √3) (2√3) / (2 - √3)(2√3)
(2√3) / (4 - 3) 2√3
Conclusion
Through these detailed methods, we have calculated the exact value of tan(5π/12). The value is either -2 - √3 (when using the angle sum identity) or 2√3 (when using the cotangent and angle subtraction method). The choice of method can vary based on the context and the specific trigonometric identities one is familiar with.