To determine the value of cos given that sin = 1/3 and tan = 0, we first need to analyze the information provided. The fact that tan = 0 indicates that the angle is either 0 degrees or 180 degrees (or equivalents in radians). At these angles, the sine and cosine values behave in a specific way:
- When the angle is 0 degrees: sin(0) = 0 and cos(0) = 1.
- When the angle is 180 degrees: sin(180) = 0 and cos(180) = -1.
However, since we know that sin = 1/3, we can infer that the scenario of an angle being 0 or 180 degrees is not possible, as sine would equal 0 in those cases. Instead, let’s think through cosine using the Pythagorean identity:
According to the identity:
sin²(θ) + cos²(θ) = 1
Plugging in the value of sin:
(1/3)² + cos²(θ) = 1
This further simplifies to:
1/9 + cos²(θ) = 1
To isolate cos²(θ), subtract 1/9 from both sides:
cos²(θ) = 1 - 1/9
cos²(θ) = 8/9
Taking the square root gives us:
cos(θ) = ±√(8/9)
cos(θ) = ±(2√2/3)
Thus, the value of cos can be either:
- cos(θ) = 2√2/3 (for angles in the first quadrant)
- cos(θ) = -2√2/3 (for angles in the second quadrant)
In summary, the value of cosine is either 2√2/3 or -2√2/3 depending on the quadrant in which the angle lies. This highlights the need to consider both positive and negative possibilities when dealing with trigonometric identities.