What are the operating signs of the tangent function in Quadrant 3 of the unit circle?

What are the operating signs of the tangent function in Quadrant 3 of the unit circle?

• A. Positive
• B. Negative
• C. Zero
• D. Undefined

Answer: (A)

To determine the operating signs of the tangent function in Quadrant 3 of the unit circle, it's important to understand the definitions of the tangent function and the positioning of angles in the unit circle. In the unit circle, tangent is defined as the ratio of the sine function to the cosine function, specifically \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). In Quadrant 3, both sine and cosine values are negative because coordinates in this quadrant have negative \(x\) (cosine) and negative \(y\) (sine) components. When both sine and cosine are negative, their ratio (the tangent function) will be positive. This is because a negative divided by a negative results in a positive value. Therefore, in Quadrant 3, the tangent function takes on positive values. This understanding clarifies why the answer suggesting that the tangent function is positive in Quadrant 3 is correct. The evaluation of signs in different quadrants is crucial for grasping trigonometric functions' behaviors, particularly when dealing with them in various mathematical contexts like graphing and solving equations.

To determine the operating signs of the tangent function in Quadrant 3 of the unit circle, it's important to understand the definitions of the tangent function and the positioning of angles in the unit circle.

In the unit circle, tangent is defined as the ratio of the sine function to the cosine function, specifically ( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} ). In Quadrant 3, both sine and cosine values are negative because coordinates in this quadrant have negative (x) (cosine) and negative (y) (sine) components.

When both sine and cosine are negative, their ratio (the tangent function) will be positive. This is because a negative divided by a negative results in a positive value. Therefore, in Quadrant 3, the tangent function takes on positive values.

This understanding clarifies why the answer suggesting that the tangent function is positive in Quadrant 3 is correct. The evaluation of signs in different quadrants is crucial for grasping trigonometric functions' behaviors, particularly when dealing with them in various mathematical contexts like graphing and solving equations.