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Q11(iv):
State whether the following are true or false. Justify your answer. (iv) $\cot A$ is the product of cot and $A$.
Solution :
Given: The statement "$\cot A$ is the product of $\cot$ and $A$".
To Find: Determine whether the given statement is True or False and provide a mathematical justification.
Step 1: Understanding Trigonometric Notation
In trigonometry, the notation $\cot A$ is a shorthand representation for the "cotangent of the angle $A$". Here, $\cot$ is not a separate algebraic variable or a numerical constant, but rather a functional operator (a trigonometric ratio) that acts upon the argument $A$.
Step 2: Analyzing the Relationship
Let us consider a right-angled triangle $\triangle ABC$ where $\angle B = 90^\circ$ and $\angle A$ is one of the acute angles. By definition, the cotangent of angle $A$ is the ratio of the length of the side adjacent to angle $A$ to the length of the side opposite to angle $A$.
Mathematically, $\cot A = \frac{\text{Adjacent side}}{\text{Opposite side}} = \frac{AB}{BC}$.
Step 3: Evaluating the "Product" Claim
If $\cot A$ were the product of $\cot$ and $A$, then $\cot$ would have to be a value that could be multiplied by $A$. However, $\cot$ by itself has no independent numerical value. It is an operator that requires an angle (the argument) to produce a ratio. If we were to separate them, the expression $\cot$ would be meaningless in the context of geometry and trigonometry.
Step 4: Logical Conclusion
Since $\cot A$ represents a single functional entity where $A$ is the angle associated with the cotangent ratio, it cannot be interpreted as the algebraic product of two distinct factors, $\cot$ and $A$. Therefore, the assertion that $\cot A$ is the product of $\cot$ and $A$ is mathematically incorrect.
Final Answer: False. The term $\cot A$ is a single trigonometric ratio representing the cotangent of angle $A$. The symbol $\cot$ is not a separate variable, and thus $\cot A$ is not the product of $\cot$ and $A$.
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) $\sin C, \cos C$
- 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) $\cos A \cos C – \sin A \sin C$
- 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(i): State whether the following are true or false. Justify your answer. (i) The value of $\tan A$ is always less than 1.
- 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(v): State whether the following are true or false. Justify your answer. (v) $\sin \theta = \frac{4}{3}$ for some angle $\theta$.
CBSE Solutions for Class 10 Mathematics Introduction to Trigonometry
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