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.