Question

which statement best explains why \\(\tan(\theta) = \tan(180^{\circ} + \theta)\\)?
tangent values are the same when angles have identical \\(x\\)-coordinates on the unit circle.
tangent values repeat when multiples of \\(180^{\circ}\\) are added to angles on the unit circle.
tangent values repeat when multiples of \\(90^{\circ}\\) are added to angles on the unit circle.
tangent values are the same when angles have identical \\(y\\)-coordinates on the unit circle.

Explanation:

To determine why \(\tan(\theta)=\tan(180^{\circ}+\theta)\), we analyze the properties of the tangent function on the unit circle. The tangent function has a period of \(180^{\circ}\) (or \(\pi\) radians), meaning \(\tan(\theta + n\cdot180^{\circ})=\tan(\theta)\) for any integer \(n\).

• The first option is incorrect because tangent is \(\frac{y}{x}\) (not related to identical \(x\)-coordinates alone).
• The second option states that tangent values repeat when multiples of \(180^{\circ}\) are added to angles. Since \(180^{\circ}\) is a multiple of \(180^{\circ}\) (with \(n = 1\) here, as we add \(180^{\circ}\) to \(\theta\)), this matches the period property of tangent.
• The third option is incorrect because the period of tangent is \(180^{\circ}\), not \(90^{\circ}\) (adding \(90^{\circ}\) changes the sign or makes it undefined in some cases).
• The fourth option is incorrect because tangent depends on the ratio \(\frac{y}{x}\), not identical \(y\)-coordinates alone.

Answer:

B. Tangent values repeat when multiples of \(180^{\circ}\) are added to angles on the unit circle.