To solve the expression [tex]\(\operatorname{cosec}^{-1}(-1) + \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)\)[/tex], we need to identify the angles that correspond to these inverse trigonometric functions and then add them together.
### Step 1: Calculate [tex]\(\operatorname{cosec}^{-1}(-1)\)[/tex]
The inverse cosecant function, [tex]\(\operatorname{cosec}^{-1}(x)\)[/tex], refers to the angle [tex]\(\theta\)[/tex] such that [tex]\(\operatorname{cosec}(\theta) = x\)[/tex].
For [tex]\(\operatorname{cosec}^{-1}(-1)\)[/tex], we need to find [tex]\(\theta\)[/tex] such that:
[tex]\[ \operatorname{cosec}(\theta) = -1 \][/tex]
The cosecant function is the reciprocal of the sine function:
[tex]\[ \operatorname{cosec}(\theta) = \frac{1}{\sin(\theta)} \][/tex]
So, we have:
[tex]\[ \frac{1}{\sin(\theta)} = -1 \implies \sin(\theta) = -1 \][/tex]
The angle [tex]\(\theta\)[/tex] that satisfies [tex]\(\sin(\theta) = -1\)[/tex] is [tex]\(\theta = -\frac{\pi}{2}\)[/tex].
Therefore,
[tex]\[ \operatorname{cosec}^{-1}(-1) = -\frac{\pi}{2} \][/tex]
### Step 2: Calculate [tex]\(\cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)\)[/tex]
The inverse cotangent function, [tex]\(\cot^{-1}(x)\)[/tex], refers to the angle [tex]\(\theta\)[/tex] such that [tex]\(\cot(\theta) = x\)[/tex].
For [tex]\(\cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)\)[/tex], we need to find [tex]\(\theta\)[/tex] such that:
[tex]\[ \cot(\theta) = -\frac{1}{\sqrt{3}} \][/tex]
The cotangent function is the reciprocal of the tangent function:
[tex]\[ \cot(\theta) = \frac{1}{\tan(\theta)} \][/tex]
So, we have:
[tex]\[ \frac{1}{\tan(\theta)} = -\frac{1}{\sqrt{3}} \implies \tan(\theta) = -\sqrt{3} \][/tex]
The angle [tex]\(\theta\)[/tex] that satisfies [tex]\(\tan(\theta) = -\sqrt{3}\)[/tex] in the principal range [tex]\([0, \pi)\)[/tex] is [tex]\(\theta = -\frac{\pi}{3}\)[/tex].
Therefore,
[tex]\[ \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{3} \][/tex]
### Step 3: Add the results
Now, we add the angles we found:
[tex]\[ \operatorname{cosec}^{-1}(-1) + \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{2} + -\frac{\pi}{3} \][/tex]
To add these fractions, we first find a common denominator. The common denominator for 2 and 3 is 6:
[tex]\[ -\frac{\pi}{2} = -\frac{3\pi}{6} \][/tex]
[tex]\[ -\frac{\pi}{3} = -\frac{2\pi}{6} \][/tex]
So, the sum is:
[tex]\[ -\frac{3\pi}{6} + -\frac{2\pi}{6} = -\frac{5\pi}{6} \][/tex]
Therefore, the result of the expression [tex]\(\operatorname{cosec}^{-1}(-1) + \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)\)[/tex] is:
[tex]\[ -\frac{5\pi}{6} \][/tex]
As numerical approximation:
[tex]\[ -\frac{5\pi}{6} \approx -2.617993877991494 \][/tex]
