Question

How do you find \[\theta \] if \[0 < \theta < 360\] and \[\cot \theta =-1\] and theta is in quadrant IV?

Answer 1

Hint: We are asked to find the value at an angle \[\theta \] such that \[\cot \theta =-1\] and the angle lies in the fourth quadrant also with 0 to 360 degrees. To answer this we need to know first that when it is negative and then using that value, we will find when will it be negative in the fourth quadrant. As we know the fourth quadrant is 0 to 360 degrees, then the angle can range from 270 degrees to 360 degrees. So, our angle lies between them.

Complete step by step answer:
We are given that \[\cot \theta \] is – 1 in the fourth quadrant. We have to find \[\theta \] such that it must be between 0 to 360 degrees. To answer this we will first see that when is \[\cot \theta \] 1 in quadrant I and then we will use the property that \[\cot \left( -\theta \right)=-\cot \theta .\] To find \[-\theta \] then we will convert this \[-\theta \] angle into the angle that will lie in the fourth quadrant. Now, we know \[\cot \theta \] is also defined at cot tangent. Both cot and tan ratio are 1 at the angle of 45 degrees. So, \[\cot {{45}^{\circ }}=1.\] Hence, we get \[\theta ={{45}^{\circ }}\] in quadrant I where \[\cot \theta \] is 1. We know \[\cot \left( -\theta \right)=-\cot \theta .\] So, as \[\cot \left( {{45}^{\circ }} \right)=1,\] so we get,
\[\cot \left( -{{45}^{\circ }} \right)=-\cot \left( {{45}^{\circ }} \right)=-1\]
Hence,
\[\cot \left( -{{45}^{\circ }} \right)=-1\]
This means at \[\theta =-{{45}^{\circ }}\] \[\cot \theta \] is 45 degrees.
Now, \[-\theta \] means that we moved \[\theta \] angle in the clockwise direction. So to find the actual angle, we will first learn using the graph.

So, moving \[-{{\theta }^{\circ }}\] in clockwise direction means going 360 degrees in anticlockwise direction. Hence, our angle will be \[{{360}^{\circ }}-\theta \] that is \[{{360}^{\circ }}-{{45}^{\circ }}.\] So, solving this we get, \[{{360}^{\circ }}-{{45}^{\circ }}={{315}^{\circ }}.\]

Note: While finding angle in any quadrant we always find the angle in the first quadrant then we will use the following relation. If angle in quadrant I is \[\theta \] then angle in quadrant II is given as \[\pi -\theta \] or \[{{180}^{\circ }}-\theta .\] Angle in quadrant III is given as \[\pi +\theta \] or \[{{180}^{\circ }}+\theta \] and lastly as done above, angle in quadrant IV is given as \[2\pi -\theta \] or \[{{360}^{\circ }}-\theta .\]