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.
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
True
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.
False
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.
False
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.
False
sin θ = Height/Hypotenuse
∴ sin θ will always less than 1 and it can never be 4/3 for any value of θ.