11. 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 = 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 θ = 4/3 for some angle θ.

(i) The value of tan A is always less than 1. Answer: False Proof: In ΔMNC in which ∠N = 90∘, MN = 3, NC = 4 and MC = 5 Value of tan M = 4/3 which is greater than. The triangle can be formed with sides equal to 3, 4 and hypotenuse = 5 as it will follow the Pythagoras theorem. MC2=MN2+NC2 52=32+42 25=9+16 25 = 25 (ii) sec A = 12/5 for some value of angle A Answer: True Justification: Let a ΔMNC in which ∠N = 90º, MC=12k and MB=5k, where k is a positive real number. By Pythagoras theorem we get, MC2=MN2+NC2 (12k)2=(5k)2+NC2 NC2+25k2=144k2 NC2=119k2 Such a triangle is possible as it will follow the Pythagoras theorem. (iii) cos A is the abbreviation used for the cosecant of angle A. Answer: False Justification: Abbreviation used for cosecant of angle M is cosec M. cos M is the abbreviation used for cosine of angle M. (iv) cot A is the product of cot and A. Answer: False Justification: cot M is not the product of cot and M. It is the cotangent of ∠M. (v) sin θ = 4/3 for some angle θ. Answer: False Justification: sin θ = Height/Hypotenuse We know that in a right angled triangle, Hypotenuse is the longest side. ∴ sin θ will always less than 1 and it can never be 4/3 for any value of θ.