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 = 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 for some angle .
False. Justification: The value of tan A is not always less than 1. For example, from the context, tan 60° = , which is approximately 1.732, and > 1. Also, tan 45° = 1.
True. Justification: sec A = 12/5 implies that in a right triangle, the ratio of the hypotenuse to the base is 12:5. Since 12 > 5, such a triangle is possible (perpendicular = = ), so an angle A exists for this value.
False. Justification: cos A is the abbreviation used for the cosine of angle A, not the cosecant. The abbreviation for cosecant of angle A is cosec A.
False. Justification: cot A is a single trigonometric ratio symbol representing the ratio of cosine to sine (or base to perpendicular). It is not a product of 'cot' and 'A'.
False. Justification: sin θ represents the ratio of perpendicular to hypotenuse. Since the perpendicular is always less than or equal to the hypotenuse in a right triangle, sin θ 1. Here, 4/3 > 1, so sin θ cannot be 4/3 for any angle θ.
Explanation
The question tests the fundamental understanding of trigonometric ratios, their abbreviations, and their valid ranges. (i) is false because tan θ can exceed 1 (e.g., tan 60°). (ii) is true because sec A is the reciprocal of cos A, and since cos A can be less than 1, sec A can be greater than 1. (iii) is false as standard abbreviations are distinct (cos vs cosec). (iv) is false because trigonometric functions are functions of angles, not algebraic products. (v) is false because the sine of an angle cannot exceed 1.