Question

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

Answer 1

(i) Consider a $ΔABC$, right-angled at B.

tan A=$\frac{12}{5}$
But$\frac{12}{5} > 1$
$∴$ tan A$> 1$
So, tan A $< 1$ is not always true.
Hence, the given statement is false.

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(ii) sec A = $\frac{12}{5}$

$\frac{AC}{AB}=\frac{12}{5}$
Let AC be 12k, AB will be 5k, where k is a positive integer.

Applying Pythagoras theorem in $ΔABC,$ we obtain
$AC^ 2 = AB ^2 + BC^ 2$
$(12k) ^2 = (5k)^ 2 + BC^ 2$
$144k ^2 = 25k ^2 + BC ^2$
$BC^ 2 = 119k^ 2$
$BC = 10.9k$
It can be observed that for given two sides AC = 12k and AB = 5k,
BC should be such that,
$AC - AB < BC < AC + AB$
$12k - 5k < BC < 12k + 5k$
$7k < BC < 17 k$

However, BC = 10.9k. Clearly, such a triangle is possible and hence, such value of sec A is possible.
Hence, the given statement is true.

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(iii) Abbreviation used for cosecant of angle A is cosec A. And cos A is the abbreviation used for cosine of angle A.
Hence, the given statement is false.

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(iv) cot A is not the product of cot and A. It is the cotangent of ∠A.
Hence, the given statement is false.

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(v) sin $θ =\frac{4}{3}$
We know that in a right-angled triangle,
$\text{sin θ} = \frac{\text{Opposite}}{\text{Hypotenuse}}$
In a right-angled triangle, hypotenuse is always greater than the remaining two sides. Therefore, such value of $\text{sin θ}$ is not possible.
Hence, the given statement is false.