Question
hat signs are sec(-140) and cot(-140)
?
?
Answers
Answered by
GPT-4o mini
To find the signs of \(\sec(-140^\circ)\) and \(\cot(-140^\circ)\), we first need to understand their respective angles in the context of the unit circle.
1. **Secant Function**:
- The secant function is defined as \(\sec(\theta) = \frac{1}{\cos(\theta)}\).
- The cosine of an angle is positive in the first and fourth quadrants and negative in the second and third quadrants.
- The angle \(-140^\circ\) can be found by adding \(360^\circ\) (to convert it to a positive angle):
\[
-140^\circ + 360^\circ = 220^\circ
\]
- The angle \(220^\circ\) is in the third quadrant, where the cosine is negative. Therefore, \(\sec(-140^\circ) = \sec(220^\circ)\) is also negative.
2. **Cotangent Function**:
- The cotangent function is defined as \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\).
- The sine function is negative in the third and fourth quadrants, while the cosine function is also negative in the third quadrant.
- Since \(-140^\circ\) corresponds to \(220^\circ\), and \(220^\circ\) is in the third quadrant, both \(\sin(220^\circ)\) and \(\cos(220^\circ)\) are negative.
- Therefore, \(\cot(-140^\circ) = \cot(220^\circ) = \frac{\cos(220^\circ)}{\sin(220^\circ)}\) will be positive (since a negative divided by a negative is positive).
In summary:
- \(\sec(-140^\circ)\) is **negative**.
- \(\cot(-140^\circ)\) is **positive**.
1. **Secant Function**:
- The secant function is defined as \(\sec(\theta) = \frac{1}{\cos(\theta)}\).
- The cosine of an angle is positive in the first and fourth quadrants and negative in the second and third quadrants.
- The angle \(-140^\circ\) can be found by adding \(360^\circ\) (to convert it to a positive angle):
\[
-140^\circ + 360^\circ = 220^\circ
\]
- The angle \(220^\circ\) is in the third quadrant, where the cosine is negative. Therefore, \(\sec(-140^\circ) = \sec(220^\circ)\) is also negative.
2. **Cotangent Function**:
- The cotangent function is defined as \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\).
- The sine function is negative in the third and fourth quadrants, while the cosine function is also negative in the third quadrant.
- Since \(-140^\circ\) corresponds to \(220^\circ\), and \(220^\circ\) is in the third quadrant, both \(\sin(220^\circ)\) and \(\cos(220^\circ)\) are negative.
- Therefore, \(\cot(-140^\circ) = \cot(220^\circ) = \frac{\cos(220^\circ)}{\sin(220^\circ)}\) will be positive (since a negative divided by a negative is positive).
In summary:
- \(\sec(-140^\circ)\) is **negative**.
- \(\cot(-140^\circ)\) is **positive**.