Question
What signs are cos(-100)
and tan(-100)
?
and tan(-100)
?
Answers
GPT-4o mini
To determine the signs of \(\cos(-100^\circ)\) and \(\tan(-100^\circ)\), we can use the properties of trigonometric functions and the unit circle.
1. **Cosine function**: The cosine function is an even function, which means that \(\cos(-x) = \cos(x)\). Therefore, \(\cos(-100^\circ) = \cos(100^\circ)\).
To find the sign of \(\cos(100^\circ)\):
- The angle \(100^\circ\) is in the second quadrant, where the cosine is negative.
Thus, \(\cos(-100^\circ) = \cos(100^\circ) < 0\).
2. **Tangent function**: The tangent function is an odd function, meaning that \(\tan(-x) = -\tan(x)\). Therefore, \(\tan(-100^\circ) = -\tan(100^\circ)\).
To find the sign of \(\tan(100^\circ)\):
- The angle \(100^\circ\) is also in the second quadrant, where the tangent is negative.
Thus, \(\tan(100^\circ) < 0\), and consequently, \(\tan(-100^\circ) = -\tan(100^\circ) > 0\).
### Summary:
- \(\cos(-100^\circ) < 0\) (negative)
- \(\tan(-100^\circ) > 0\) (positive)
1. **Cosine function**: The cosine function is an even function, which means that \(\cos(-x) = \cos(x)\). Therefore, \(\cos(-100^\circ) = \cos(100^\circ)\).
To find the sign of \(\cos(100^\circ)\):
- The angle \(100^\circ\) is in the second quadrant, where the cosine is negative.
Thus, \(\cos(-100^\circ) = \cos(100^\circ) < 0\).
2. **Tangent function**: The tangent function is an odd function, meaning that \(\tan(-x) = -\tan(x)\). Therefore, \(\tan(-100^\circ) = -\tan(100^\circ)\).
To find the sign of \(\tan(100^\circ)\):
- The angle \(100^\circ\) is also in the second quadrant, where the tangent is negative.
Thus, \(\tan(100^\circ) < 0\), and consequently, \(\tan(-100^\circ) = -\tan(100^\circ) > 0\).
### Summary:
- \(\cos(-100^\circ) < 0\) (negative)
- \(\tan(-100^\circ) > 0\) (positive)