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) Let \(ΔABC\) be a right-angled triangle at B. tan A=\(\frac{12}{5}\) Since \(\frac{12}{5} > 1 \), it follows that tan A \( > 1\). Therefore, the statement "tan A \( < 1\) is always true" is false. The given statement is false.

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(ii) Given sec A =\(\frac{12}{5}\). This implies \(\frac{AC}{AB}=\frac{12}{5}\). Let AC = 12k and AB = 5k, where k is a positive integer. Using the Pythagorean theorem in \(ΔABC\): \(AC^ 2 = AB ^2 + BC^ 2 \) \( (12k) ^2 = (5k)^ 2 + BC^ 2 \) \( 144k^ 2 = 25k^ 2 + BC^ 2 \) \( BC^ 2 = 119k^ 2\) \( BC = \sqrt{119}k \approx 10.9k \) For any triangle, the sum of two sides must be greater than the third side, and the difference between two sides must be less than the third side. Thus, \( AC - AB < BC < AC + AB \), which means \( 7k < BC < 17 k \). Since \( BC \approx 10.9k \) falls within this range, such a triangle is possible, and therefore, such a value of sec A is possible. The given statement is true.

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(iii) The abbreviation for cosecant of angle A is cosec A. The abbreviation for cosine of angle A is cos A. The given statement is false.

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(iv) cot A represents the cotangent of angle A, not the product of 'cot' and 'A'. The given statement is false.

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(v) Given sin \(θ =\frac{4}{3}\). In a right-angled triangle, sin θ = \(\frac{\text{Opposite}}{\text{Hypotenuse}}\). The hypotenuse is always the longest side in a right-angled triangle. Therefore, sin θ cannot be greater than 1. Hence, the value sin \(θ =\frac{4}{3}\) is not possible. The given statement is false.