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Q6:
Find the principal values of the following: $\tan^{-1} (-1)$
Solution :
Step 1: Definition of the Principal Value Branch
To find the principal value of $y = \tan^{-1}(-1)$, we must refer to the range of the principal value branch of the inverse tangent function. The domain of the function $f(x) = \tan^{-1}(x)$ is $\mathbb{R}$, and its range is the open interval $(-\frac{\pi}{2}, \frac{\pi}{2})$.
Thus, we seek an angle $y$ such that:
$\tan(y) = -1$
where $y \in (-\frac{\pi}{2}, \frac{\pi}{2})$ [By Definition of Principal Value Branches].
Step 2: Evaluating the Trigonometric Identity
We know from the standard trigonometric table that $\tan(\frac{\pi}{4}) = 1$. Utilizing the property of odd functions for the tangent ratio, specifically $\tan(-\theta) = -\tan(\theta)$ [Per the property of odd functions in trigonometry], we can express the negative value as follows:
$\tan(-y) = -\tan(y)$
Substitute $\theta = \frac{\pi}{4}$ into the identity:
$\tan\left(-\frac{\pi}{4}\right) = -\tan\left(\frac{\pi}{4}\right)$
Since $\tan\left(\frac{\pi}{4}\right) = 1$, it follows that:
$\tan\left(-\frac{\pi}{4}\right) = -1$
Step 3: Verification of the Range Constraint
The value obtained is $y = -\frac{\pi}{4}$. We must verify if this value lies within the defined range $(-\frac{\pi}{2}, \frac{\pi}{2})$.
- Lower bound: $-\frac{\pi}{2} \approx -1.5707$ radians
- Upper bound: $\frac{\pi}{2} \approx 1.5707$ radians
- Calculated value: $-\frac{\pi}{4} \approx -0.7854$ radians
Since $-\frac{\pi}{2} < -\frac{\pi}{4} < \frac{\pi}{2}$, the value $-\frac{\pi}{4}$ is indeed the principal value [By the definition of the inverse tangent range].
Step 4: Geometric Representation of the Inverse Tangent
Final Solution: The principal value of $\tan^{-1}(-1)$ is $-\frac{\pi}{4}$.
More Questions from Class 12 Mathematics Inverse Trigonometric Functions EXERCISE 2.1
- Q1: Find the principal values of the following: $\sin^{-1} (\frac{-1}{2})$
- Q2: Find the principal values of the following: $\cos^{-1} (\frac{\sqrt{3}}{2})$
- Q3: Find the principal values of the following: $\csc^{-1} (2)$
- Q4: Find the principal values of the following: $\tan^{-1} (-\sqrt{3})$
- Q5: Find the principal values of the following: $\cos^{-1} (\frac{-1}{2})$
- Q7: Find the principal values of the following: $\sec^{-1} (\frac{2}{\sqrt{3}})$
- Q8: Find the principal values of the following: $\cot^{-1} (\sqrt{3})$
- Q9: Find the principal values of the following: $\cos^{-1} (\frac{-1}{\sqrt{2}})$
- Q10: Find the principal values of the following: $\cosec^{-1} (-\sqrt{2})$
- Q11: Find the values of the following: $\tan^{-1}(1) + \cos^{-1}(\frac{-1}{2}) + \sin^{-1}(\frac{-1}{2})$
- Q12: Find the values of the following: $\cos^{-1}(\frac{1}{2}) + 2\sin^{-1}(\frac{1}{2})$
- Q13: If $\sin^{-1} x = y$, then
- Q14: $\tan^{-1}(\sqrt{3}) - \sec^{-1}(-2)$ is equal to
CBSE Solutions for Class 12 Mathematics Inverse Trigonometric Functions
Chapters in CBSE - Class 12 Mathematics
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