Find the principal values of the following :
`tan^(-1) (1) +cos^(-1) (-1/2) +sin^(-1) (-1/2)`

Answer

To find the principal values of the expression \( \tan^{-1}(1) + \cos^{-1}(-\frac{1}{2}) + \sin^{-1}(-\frac{1}{2}) \), we will evaluate each term step by step.

### Step 1: Evaluate \( \tan^{-1}(1) \)

The value of \( \tan^{-1}(1) \) corresponds to the angle whose tangent is 1. The principal value of \( \tan^{-1}(1) \) is:

\[ \tan^{-1}(1) = \frac{\pi}{4} \]

### Step 2: Evaluate \( \cos^{-1}(-\frac{1}{2}) \)

The value of \( \cos^{-1}(-\frac{1}{2}) \) corresponds to the angle whose cosine is \(-\frac{1}{2}\). The principal value of \( \cos^{-1}(-\frac{1}{2}) \) is:

\[ \cos^{-1}(-\frac{1}{2}) = \frac{2\pi}{3} \]

### Step 3: Evaluate \( \sin^{-1}(-\frac{1}{2}) \)

The value of \( \sin^{-1}(-\frac{1}{2}) \) corresponds to the angle whose sine is \(-\frac{1}{2}\). The principal value of \( \sin^{-1}(-\frac{1}{2}) \) is:

\[ \sin^{-1}(-\frac{1}{2}) = -\frac{\pi}{6} \]

### Step 4: Combine the values

Now we can combine the values we found:

\[ \tan^{-1}(1) + \cos^{-1}(-\frac{1}{2}) + \sin^{-1}(-\frac{1}{2}) = \frac{\pi}{4} + \frac{2\pi}{3} - \frac{\pi}{6} \]

### Step 5: Find a common denominator

The common denominator for \(4\), \(3\), and \(6\) is \(12\). We will convert each term:

\[ \frac{\pi}{4} = \frac{3\pi}{12}, \quad \frac{2\pi}{3} = \frac{8\pi}{12}, \quad -\frac{\pi}{6} = -\frac{2\pi}{12} \]

### Step 6: Add the fractions

Now we can add the fractions:

\[ \frac{3\pi}{12} + \frac{8\pi}{12} - \frac{2\pi}{12} = \frac{3\pi + 8\pi - 2\pi}{12} = \frac{9\pi}{12} \]

### Step 7: Simplify the result

Finally, we simplify \( \frac{9\pi}{12} \):

\[ \frac{9\pi}{12} = \frac{3\pi}{4} \]

### Final Answer

Thus, the principal value of the expression \( \tan^{-1}(1) + \cos^{-1}(-\frac{1}{2}) + \sin^{-1}(-\frac{1}{2}) \) is:

\[ \frac{3\pi}{4} \]

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