what is tan(-5π/6)-cot(7Pπ/2) without a calculator
The Correct Answer and Explanation is :
Let’s evaluate the expression ( \tan \left( -\frac{5\pi}{6} \right) – \cot \left( \frac{7\pi}{2} \right) ) step by step.
Step 1: Evaluate ( \tan \left( -\frac{5\pi}{6} \right) )
We start by simplifying the tangent term. The tangent function has a periodicity of ( \pi ), meaning that ( \tan(\theta) = \tan(\theta + n\pi) ), where ( n ) is any integer.
Finding a Coterminal Angle
First, we find a coterminal angle for ( -\frac{5\pi}{6} ) by adding ( 2\pi ) (since ( 2\pi ) is one full revolution):
[
-\frac{5\pi}{6} + 2\pi = -\frac{5\pi}{6} + \frac{12\pi}{6} = \frac{7\pi}{6}
]
Now, we know that ( \tan \left( -\frac{5\pi}{6} \right) = \tan \left( \frac{7\pi}{6} \right) ), since the tangent function has a period of ( \pi ).
Evaluating ( \tan \left( \frac{7\pi}{6} \right) )
The angle ( \frac{7\pi}{6} ) is in the third quadrant (since ( \frac{7\pi}{6} > \pi ), but less than ( \frac{3\pi}{2} )), where the tangent function is positive. The reference angle is:
[
\frac{7\pi}{6} – \pi = \frac{\pi}{6}
]
Since ( \tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}} ), we have:
[
\tan \left( \frac{7\pi}{6} \right) = \frac{1}{\sqrt{3}}
]
Thus, ( \tan \left( -\frac{5\pi}{6} \right) = \frac{1}{\sqrt{3}} ).
Step 2: Evaluate ( \cot \left( \frac{7\pi}{2} \right) )
The cotangent function has a periodicity of ( \pi ), meaning that ( \cot(\theta) = \cot(\theta + n\pi) ).
Finding a Coterminal Angle
Now, we simplify ( \frac{7\pi}{2} ) by subtracting ( 2\pi ) (which is equivalent to subtracting ( \frac{4\pi}{2} )) to bring it within one full cycle:
[
\frac{7\pi}{2} – 2\pi = \frac{7\pi}{2} – \frac{4\pi}{2} = \frac{3\pi}{2}
]
Thus, ( \cot \left( \frac{7\pi}{2} \right) = \cot \left( \frac{3\pi}{2} \right) ).
Evaluating ( \cot \left( \frac{3\pi}{2} \right) )
The angle ( \frac{3\pi}{2} ) lies on the negative ( y )-axis. Since cotangent is the reciprocal of tangent, and ( \tan \left( \frac{3\pi}{2} \right) = 0 ), we have:
[
\cot \left( \frac{3\pi}{2} \right) = \frac{1}{0}
]
This is undefined.
Conclusion
Since ( \cot \left( \frac{7\pi}{2} \right) ) is undefined, the entire expression ( \tan \left( -\frac{5\pi}{6} \right) – \cot \left( \frac{7\pi}{2} \right) ) is undefined.
Thus, the correct answer is that the expression is undefined.