To solve for the value of Cos(α) given that Sin(α) = 1/4 and Tan(α) = 0, we can follow these steps:
- Understanding Tan(α):
The tangent of an angle α is defined as the ratio of the sine to cosine:
Tan(α) = Sin(α) / Cos(α)
If Tan(α) = 0, it implies that Sin(α) = 0 or Cos(α) is undefined. However, in this case, we know that Sin(α) = 1/4, which means Cos(α) must be undefined. Hence, this would dictate that α is either at 0 degrees or 180 degrees since those angles yield a tangent of 0, where the sine value can still be defined. - Using Pythagorean Identity:
Regardless, we can still derive values using the Pythagorean identity for sine and cosine:
Sin²(α) + Cos²(α) = 1
Plugging in the known value of sine:
(1/4)² + Cos²(α) = 1
1/16 + Cos²(α) = 1 - Solving for Cos(α):
Rearranging gives us:
Cos²(α) = 1 - 1/16
Cos²(α) = 16/16 - 1/16
Cos²(α) = 15/16
Taking the square root of both sides:
Cos(α) = ±√(15/16)
This simplifies to:
Cos(α) = ±(√15)/4
Therefore, the possible values for Cos(α) are (√15)/4 or -(√15)/4, depending on the quadrant in which the angle α lies.