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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 $A$.

Solution :

Given: The statement "$\cos A$ is the abbreviation used for the cosecant of angle $A$."

To Find: Determine whether the given statement is True or False and provide a mathematical justification.

Step 1: Defining Trigonometric Abbreviations

In trigonometry, the six primary trigonometric functions are defined by specific abbreviations derived from the names of the ratios of sides in a right-angled triangle. Let us define them formally:

  • $\sin A$: Abbreviation for sine of angle $A$.
  • $\cos A$: Abbreviation for cosine of angle $A$.
  • $\tan A$: Abbreviation for tangent of angle $A$.
  • $\cot A$: Abbreviation for cotangent of angle $A$.
  • $\sec A$: Abbreviation for secant of angle $A$.
  • $\csc A$ (or $\text{cosec } A$): Abbreviation for cosecant of angle $A$.

Step 2: Analyzing the Statement

The statement claims that $\cos A$ is the abbreviation for the cosecant of angle $A$. Based on the definitions established in Step 1:

1. The abbreviation $\cos A$ corresponds to the cosine of angle $A$.

2. The abbreviation for the cosecant of angle $A$ is $\csc A$ or $\text{cosec } A$.

Step 3: Logical Comparison

Since $\cos A$ represents the cosine function and not the cosecant function, the statement provided is mathematically incorrect. The term "cosecant" is the reciprocal of the sine function ($\csc A = \frac{1}{\sin A}$), whereas "cosine" is a distinct trigonometric ratio ($\cos A = \frac{\text{adjacent side}}{\text{hypotenuse}}$).

Conclusion:

The statement is False because $\cos A$ is the abbreviation for the cosine of angle $A$, while the abbreviation for the cosecant of angle $A$ is $\text{cosec } A$.

Final Answer: False. $\cos A$ is the abbreviation for the cosine of angle $A$, whereas the abbreviation for the cosecant of angle $A$ is $\text{cosec } A$.


More Questions from Class 10 Mathematics Introduction to Trigonometry EXERCISE 8.1


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