Final answer:
To find the exact value of tangent (19pi/12), we can find the reference angle and use the tangent formula. The exact value is √23 / 23.
Step-by-step explanation:
To determine the exact value of Σ (19π / 12), we need to find the tangent of that angle. The key to finding the exact value is to relate the given angle to a reference angle within the unit circle. First, we divide 19π / 12 by 2 to get 19π / 24. This brings us to the 1st Quadrant in the unit circle, meaning that the tangent value will be positive since both sine and cosine are positive in the 1st Quadrant.
Now, we find the reference angle by subtracting a full revolution (2π) from the given angle: 19π / 24 - 2π = -5π / 24. The tangent function repeats itself every 180 degrees (or π), so we can adjust the reference angle to a positive value by adding 2π: -5π / 24 + 2π = 3π / 24.
Now, we can take the tangent of the reference angle: tan(3π / 24) = sin(3π / 24) / cos(3π / 24). Using the values from the unit circle, sin(3π / 24) = 1/8 and cos(3π / 24) = √23 / 8. Plugging these values into the tangent formula, we get tan(3π / 24) = (1/8) / (√23 / 8) = (1/8) * (8 / √23) = 1 / √23 = √23 / 23