Question
Describe how to determine the sign for tan0 in each quadrant of the coordinate plane
Answers
Answered by
GPT-4o mini
To determine the sign of the tangent function (tan θ) in each quadrant of the coordinate plane, it is important to understand the definitions of the trigonometric functions based on the coordinates of points on the unit circle:
1. **Quadrant I (0 to 90 degrees or 0 to π/2 radians)**:
- In this quadrant, both the x-coordinate (cos θ) and the y-coordinate (sin θ) are positive.
- Since tangent is defined as tan θ = sin θ / cos θ, the ratio of two positive values is positive.
- **Sign of tan θ: Positive**
2. **Quadrant II (90 to 180 degrees or π/2 to π radians)**:
- In this quadrant, the x-coordinate (cos θ) is negative, and the y-coordinate (sin θ) is positive.
- Therefore, the ratio tan θ = sin θ / cos θ will have a positive numerator and a negative denominator, resulting in a negative value.
- **Sign of tan θ: Negative**
3. **Quadrant III (180 to 270 degrees or π to 3π/2 radians)**:
- Here, both the x-coordinate (cos θ) and the y-coordinate (sin θ) are negative.
- The ratio tan θ = sin θ / cos θ has both numerator and denominator negative, resulting in a positive value.
- **Sign of tan θ: Positive**
4. **Quadrant IV (270 to 360 degrees or 3π/2 to 2π radians)**:
- In this quadrant, the x-coordinate (cos θ) is positive, and the y-coordinate (sin θ) is negative.
- Thus, tan θ = sin θ / cos θ has a negative numerator and a positive denominator, resulting in a negative value.
- **Sign of tan θ: Negative**
### Summary
- **Quadrant I**: tan θ is **Positive**
- **Quadrant II**: tan θ is **Negative**
- **Quadrant III**: tan θ is **Positive**
- **Quadrant IV**: tan θ is **Negative**
This cyclical pattern of signs can be remembered by the acronym "All Students Take Calculus," which indicates:
- **A**: All (Positive in Quadrant I)
- **S**: Sine (Positive in Quadrant II)
- **T**: Tangent (Positive in Quadrant III)
- **C**: Cosine (Positive in Quadrant IV)
1. **Quadrant I (0 to 90 degrees or 0 to π/2 radians)**:
- In this quadrant, both the x-coordinate (cos θ) and the y-coordinate (sin θ) are positive.
- Since tangent is defined as tan θ = sin θ / cos θ, the ratio of two positive values is positive.
- **Sign of tan θ: Positive**
2. **Quadrant II (90 to 180 degrees or π/2 to π radians)**:
- In this quadrant, the x-coordinate (cos θ) is negative, and the y-coordinate (sin θ) is positive.
- Therefore, the ratio tan θ = sin θ / cos θ will have a positive numerator and a negative denominator, resulting in a negative value.
- **Sign of tan θ: Negative**
3. **Quadrant III (180 to 270 degrees or π to 3π/2 radians)**:
- Here, both the x-coordinate (cos θ) and the y-coordinate (sin θ) are negative.
- The ratio tan θ = sin θ / cos θ has both numerator and denominator negative, resulting in a positive value.
- **Sign of tan θ: Positive**
4. **Quadrant IV (270 to 360 degrees or 3π/2 to 2π radians)**:
- In this quadrant, the x-coordinate (cos θ) is positive, and the y-coordinate (sin θ) is negative.
- Thus, tan θ = sin θ / cos θ has a negative numerator and a positive denominator, resulting in a negative value.
- **Sign of tan θ: Negative**
### Summary
- **Quadrant I**: tan θ is **Positive**
- **Quadrant II**: tan θ is **Negative**
- **Quadrant III**: tan θ is **Positive**
- **Quadrant IV**: tan θ is **Negative**
This cyclical pattern of signs can be remembered by the acronym "All Students Take Calculus," which indicates:
- **A**: All (Positive in Quadrant I)
- **S**: Sine (Positive in Quadrant II)
- **T**: Tangent (Positive in Quadrant III)
- **C**: Cosine (Positive in Quadrant IV)