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Q11(i):
State whether the following are true or false. Justify your answer. (i) The value of $\tan A$ is always less than 1.
Solution :
Given: A statement regarding the trigonometric ratio $\tan A$, where $A$ is an acute angle in a right-angled triangle.
To Find/Prove: Determine whether the statement "The value of $\tan A$ is always less than 1" is True or False, and provide a mathematical justification.
Visual Representation:
Step 1: Definition of Tangent Ratio
In a right-angled triangle $ABC$ right-angled at $B$, the tangent of angle $A$ is defined as the ratio of the length of the side opposite to angle $A$ to the length of the side adjacent to angle $A$.
Mathematically: $\tan A = \frac{\text{Opposite side}}{\text{Adjacent side}} = \frac{BC}{AB}$
Step 2: Analyzing the Ratio
In a right-angled triangle, the lengths of the sides $BC$ (opposite) and $AB$ (adjacent) are independent of each other, provided they are both positive real numbers. There is no geometric constraint that forces the opposite side to be smaller than the adjacent side.
Step 3: Counter-Example Verification
Let us consider a specific case where the opposite side is greater than the adjacent side. Suppose $BC = 4$ units and $AB = 3$ units.
Then, $\tan A = \frac{4}{3}$
Performing the division: $\frac{4}{3} = 1.333...$
Since $1.333... > 1$, we have found a case where $\tan A$ is greater than 1.
Step 4: Theoretical Justification
The tangent function $\tan \theta$ is defined for $0^\circ \le \theta < 90^\circ$. As $\theta$ approaches $90^\circ$, the value of $\tan \theta$ increases without bound (tends to infinity). For example, $\tan 60^\circ = \sqrt{3} \approx 1.732$, which is clearly greater than 1.
Conclusion:
Since there exist values of $A$ for which $\tan A > 1$, the statement "The value of $\tan A$ is always less than 1" is incorrect.
Final Answer: False. The value of $\tan A$ can take any real value. For instance, if $\angle A = 60^\circ$, then $\tan 60^\circ = \sqrt{3} \approx 1.732$, which is greater than 1.
More Questions from Class 10 Mathematics Introduction to Trigonometry EXERCISE 8.1
- Q1(i): In $\triangle ABC$, right-angled at $B$, $AB = 24$ cm, $BC = 7$ cm. Determine : (i) $\sin A, \cos A$
- Q1(ii): In $\triangle ABC$, right-angled at $B$, $AB = 24$ cm, $BC = 7$ cm. Determine : (ii) $\sin C, \cos C$
- Q10: In $\triangle PQR$, right-angled at $Q$, $PR + QR = 25$ cm and $PQ = 5$ cm. Determine the values of $\sin P, \cos P$ and $\tan P$.
- Q11(ii): State whether the following are true or false. Justify your answer. (ii) $\sec A = \frac{12}{5}$ for some value of angle $A$.
- Q11(iii): State whether the following are true or false. Justify your answer. (iii) $\cos A$ is the abbreviation used for the cosecant of angle $A$.
- Q11(iv): State whether the following are true or false. Justify your answer. (iv) $\cot A$ is the product of cot and $A$.
- Q11(v): State whether the following are true or false. Justify your answer. (v) $\sin \theta = \frac{4}{3}$ for some angle $\theta$.
- Q2: In Fig. 8.13, find $\tan P – \cot R$.
- Q3: If $\sin A = \frac{3}{4}$, calculate $\cos A$ and $\tan A$.
- Q4: Given $15 \cot A = 8$, find $\sin A$ and $\sec A$.
- Q5: Given $\sec \theta = \frac{13}{12}$, calculate all other trigonometric ratios.
- Q6: If $\angle A$ and $\angle B$ are acute angles such that $\cos A = \cos B$, then show that $\angle A = \angle B$.
- Q7(i): If $\cot \theta = \frac{7}{8}$, evaluate : (i) $\frac{(1 + \sin \theta) (1 - \sin \theta)}{(1 + \cos \theta) (1 - \cos \theta)}$
- Q7(ii): If $\cot \theta = \frac{7}{8}$, evaluate : (ii) $\cot^2 \theta$
- Q8: If $3 \cot A = 4$, check whether $\frac{1 - \tan^2 A}{1 + \tan^2 A} = \cos^2 A – \sin^2 A$ or not.
- Q9(i): In triangle ABC, right-angled at B, if $\tan A = \frac{1}{\sqrt{3}}$, find the value of: (i) $\sin A \cos C + \cos A \sin C$
- Q9(ii): In triangle ABC, right-angled at B, if $\tan A = \frac{1}{\sqrt{3}}$, find the value of: (ii) $\cos A \cos C – \sin A \sin C$
CBSE Solutions for Class 10 Mathematics Introduction to Trigonometry
Chapters in CBSE - Class 10 Mathematics
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