Question

State whether the following are true or false. Justify your answer.
(i) The value of $\tan {\text{A}}$ is always less than $1$.
(ii) $\sec {\text{A}} = \dfrac{{12}}{5}$ for the same value of angle ${\text{A}}$.
(iii) $\cos {\text{A}}$ is the abbreviation used for the cosecant of angle ${\text{A}}$.
(iv) $\cot {\text{A}}$ is the product of $\cot$ and ${\text{A}}$.
(v) $\sin \theta = \dfrac{4}{3}$ for some angle $\theta$.

Answer 1

For first question we will take a counter example where the value of $\tan {\text{A}}$ is greater than $1\,.$ For second question we need to have idea about the range of $\sec {\text{A}}$ i.e. what is the range in which $\sec {\text{A}}$ lies. For the third question we just need the full form $\cos {\text{A}}$ . Similarly, for fourth we need full form . For fifth question we should have the idea about the range of $\sin.$

Complete step-by-step solution:
(i) Answer: The statement is false. Reason: Let’s assume angle ${\text{A}}$ is $60^\circ$. Therefore, we can write $\tan {\text{A}} = \tan 60^\circ = \sqrt 3$. We know that $\sqrt 3 = 1.73$ which is greater than $1$.
Hence, we can say that the statement is false.
(ii) Answer: The statement is true. Reason: We know that the value of $\sec \theta$ is either greater than $1$ or less than $- 1$. Now, from the given values in the question we can write $\sec {\text{A}} = \dfrac{{12}}{5} = 2.4$. Hence, $\sec {\text{A}}$ is greater than $1$.
Therefore, the statement is true.
(iii) Answer: The statement is false. Reason: we know that $\cos {\text{A}}$ is the abbreviation used for cosine of angle ${\text{A}}$.
Hence, the given statement is false.
(iv) Answer: The statement is false. Reason: we know that $\cot {\text{A}}$ is the abbreviation used for cotangent of angle ${\text{A}}$ and also $\cot {\text{A}} = \dfrac{{{\text{Adjacent side}}}}{{{\text{opposite side}}}}$.
Hence, the given statement is false.
(v) Answer: The statement is false. Reason: we know that value of $\sin \theta$ can be $1$, $- 1$ or any value between $1$ and $- 1$ but here in the question $\sin \theta = \dfrac{4}{3} = 1.333$ which is greater than $1$.
Hence, the given statement is false.

Note: The other important things are the formula of $\sin$ , $\cos$ and $\tan$ which we need to memorize.
$\sin {\text{A = }}\dfrac{{{\text{Opposite}}}}{{{\text{Hypotenuse}}}}$
$\cos {\text{A = }}\dfrac{{{\text{Adjacent}}}}{{{\text{Hypotenuse}}}}$
$\tan {\text{A = }}\dfrac{{{\text{Opposite}}}}{{{\text{Adjacent}}}}$
$\cos ec {\text{A = }}\dfrac{{{\text{Hypotenuse}}}}{{{\text{Opposite}}}}$
$\sec {\text{A = }}\dfrac{{{\text{Hypotenuse}}}}{{{\text{Adjacent}}}}$
$\cot {\text{A = }}\dfrac{{{\text{Adjacent}}}}{{{\text{Opposite}}}}$