To find the value of cos(θ) given that sin(θ) = 1/3 and tan(θ) = 0, we’ll start by dissecting the provided information.
The equation tan(θ) = 0 implies that θ is at a position in the unit circle where the y-coordinate (which corresponds to sin(θ)) is zero. This typically occurs at θ = 0 or θ = π (or any integer multiple of π). At these angles, the sine function indeed equals zero, leading us to conclude that they do not match the given condition of sin(θ) being 1/3.
On the other hand, the value of sin(θ) = 1/3 indicates that the sine is a positive value. This situation occurs within the first and second quadrants of the unit circle. Therefore, for this specific case, tan(θ) cannot be zero since tan(θ) is defined as sin(θ)/cos(θ) and would not yield 0 if sin(θ) is positive.
Even with this contradiction, we can still calculate cos(θ). To do this, we can use the Pythagorean identity:
sin²(θ) + cos²(θ) = 1
Substituting our known value:
(1/3)² + cos²(θ) = 1
This simplifies to:
1/9 + cos²(θ) = 1
Next, we solve for cos²(θ):
cos²(θ) = 1 – 1/9
cos²(θ) = 8/9
Taking the square root of both sides, we find:
cos(θ) = ±√(8/9)
Thus, cos(θ) = ±(2√2)/3.
In conclusion, the value of cos(θ) is ±(2√2)/3, but it’s essential to note that this solution does not coincide with the condition that tan(θ) = 0, indicating that there might have been an error or misinterpretation in the initial question.