Question

is this positive if yes please tell how ?
\[\sin(\cos^{-1} (\cot 4))\]

Answer 1

@ganeshie8 ???

Answer 2

@sidsiddhartha ??

Answer 3

ok so we have to work from inside of the brackets out. what does cot4 equal to?

Answer 4

clue: cot=1/tan

Answer 5

Let's assume we're measuring angles in radians. Recall that the division of the quadrants of the unit circle:

Now, \(\pi\approx 3.14\) and \(\dfrac{3\pi}{2}\approx4.71\), so \(\theta=4\) lies in the third quadrant.

In this quadrant, angles will have negative values of sine and cosine.
\[\cot4=\frac{\cos4}{\sin4}=\frac{\text{negative}}{\text{negative}}=\text{positive}\]
If we let \(\theta=\cos^{-1}(\cot4)\), then \(\cos\theta=\cot4>0\), which tells us that \(\theta\) must be in the first. This is because the inverse cosine function has its domain in the first and second quadrants only; Q1 has the angles with positive cosines, and Q2 has those with negative cosines - we're given an angle with a positive cosine, so we choose Q1.

Any angle in the first quadrant will have positive values of sine and cosine, so \(\sin\theta>0\).