Question
State whether the following are true or false. Justify your answer.
$\sec\text{A}=\frac{12}{5}$ for some value of angle A.
$\sec\text{A}=\frac{12}{5}$ for some value of angle A.
✓
Answer
$\sec\text{A}=\frac{1}{\cos\text{A}}$
In sec A and $\cos\text{A},\angle\text{A}$ is acute angle
Therefore,
Minimum value $\angle\text{A}$ of is 0° and
Maximum value of $\angle\text{A}$ is 90°
We know that cos 0° = 1 and
cos 90° = 0
Now,
$\sec0^\circ=\frac{1}{\cos0^\circ}$
$=\frac{1}{1}$
Therefore minimum value of sec A is sec 0 ° …… (1)
Now,
$\sec90^\circ=\frac{1}{\cos90^\circ}$
$=\frac{1}{0}$
$=\infty$
Therefore maximum value of sec A is $\sec90^\circ=\infty\ \dots(2)$
Now consider the given value
$\sec\text{A}=\frac{12}{5}$
Here, $\frac{12}{5}=2.4$
This value 2.4 lies in between 1 and $\infty$
Now from equation (1) and (2), we can say that the value $\frac{12}{5}=2.4$ lies in between minimum value of sec A (that is 1) and maximum value of sec A (that is $\infty$)
Hence $\sec\text{A}=\frac{12}{5},$ for some value of angle A is true.
In sec A and $\cos\text{A},\angle\text{A}$ is acute angle
Therefore,
Minimum value $\angle\text{A}$ of is 0° and
Maximum value of $\angle\text{A}$ is 90°
We know that cos 0° = 1 and
cos 90° = 0
Now,
$\sec0^\circ=\frac{1}{\cos0^\circ}$
$=\frac{1}{1}$
Therefore minimum value of sec A is sec 0 ° …… (1)
Now,
$\sec90^\circ=\frac{1}{\cos90^\circ}$
$=\frac{1}{0}$
$=\infty$
Therefore maximum value of sec A is $\sec90^\circ=\infty\ \dots(2)$
Now consider the given value
$\sec\text{A}=\frac{12}{5}$
Here, $\frac{12}{5}=2.4$
This value 2.4 lies in between 1 and $\infty$
Now from equation (1) and (2), we can say that the value $\frac{12}{5}=2.4$ lies in between minimum value of sec A (that is 1) and maximum value of sec A (that is $\infty$)
Hence $\sec\text{A}=\frac{12}{5},$ for some value of angle A is true.
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