State whether the following are true or false. Justify your answer. (i) The value of tan A is always less than 1. (ii) sec A = 12 5 for some value of angle A. (iii) cos A is the abbreviation used for the cosecant of angle A. (iv) cot A is the product of cot and A. (v) sin = 4 3 for some angle .
Step-by-Step Solution
Key Concept: Use the definitions of the six trigonometric ratios for an acute angle $\theta$ in a right‑angled triangle: \[\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta=\frac{\text{opposite}}{\text{adjacent}};\] \[\sec\theta=\frac{1}{\cos\theta},\quad \csc\theta=\frac{1}{\sin\theta},\quad \cot\theta=\frac{1}{\tan\theta}.\] The values of \(\sin\theta\) and \(\cos\theta\) lie in $[0,1]$ for $0^\circ\le\theta\le90^\circ$, hence \(\tan\theta$ and $\cot\theta$ can be less than, equal to, or greater than 1. The reciprocal functions $\sec\theta$ and $\csc\theta$ satisfy $|\sec\theta|\ge1$ and $|\csc\theta|\ge1$. Also note the standard abbreviations: "cos" stands for cosine, not cosecant.
1. Statement (i): "The value of $\tan A$ is always less than 1."
- From the definition $\tan A = \frac{\text{opposite}}{\text{adjacent}}$, the ratio can be any non‑negative real number depending on the relative lengths of the sides. For example, if the opposite side is longer than the adjacent side, $\tan A > 1$ (e.g., $A=60^\circ$, $\tan 60^\circ = \sqrt{3} \approx 1.73$).
- Conclusion: False.
2. Statement (ii): "$\sec A = \frac{12}{5}$ for some value of angle $A$."
- $\sec A = \frac{1}{\cos A}$, so $\cos A = \frac{5}{12}$. Since $0 < \frac{5}{12} < 1$, such a cosine value is permissible for an acute angle. Indeed, $\cos^{-1}\left(\frac{5}{12}\right) \approx 65.38^\circ$.
- Conclusion: True.
3. Statement (iii): "cos A is the abbreviation used for the cosecant of angle A."
- The standard abbreviation "cos" stands for cosine, defined as $\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}$. The abbreviation for cosecant is "csc" (or sometimes "cosec").
- Conclusion: False.
4. Statement (iv): "cot A is the product of cot and A."
- The notation $\cot A$ denotes the cotangent of angle $A$, i.e., $\cot A = \frac{1}{\tan A} = \frac{\text{adjacent}}{\text{opposite}}$. It is not a product of two separate symbols "cot" and "A".
- Conclusion: False.
5. Statement (v): "$\sin \theta = \frac{4}{3}$ for some angle $\theta$."
- For any real angle, $\sin \theta$ must satisfy $-1 \le \sin \theta \le 1$. The value $\frac{4}{3} \approx 1.33$ lies outside this interval, hence no angle can have such a sine value.
- Conclusion: False.
Overall answer: (i) False, (ii) True, (iii) False, (iv) False, (v) False.
Correct Answer: (i) False, (ii) True, (iii) False, (iv) False, (v) False