State whether the following are true or false. Justify your answer

(i) Sin (A + B) = sin A + sin B


(ii) The value of sin θ increases as θ increases


(iii) The value of cos θ increases as θ increases.


(iv) Sin θ = cos θ for all values of θ.


(v) Cot A is not defined for A = 0°

(i) sin (A + B) = sin A + sin B

Let A = 30° and B = 60°


sin (A + B) = sin (30° + 60°)


= sin 90°


= 1


sin A + sin B = sin 30° + sin 60°


=


=


Clearly, sin (A + B) ≠sin A + sin B


Hence, the given statement is false


(ii) The value of sin θ increases as θ increases in the interval of 0° < θ < 90° as sin 0° = 0


sin 30o =


sin 45o = = 0.707


sin 60o = = 0.886


sin 90° = 1


Hence, the given statement is true


(iii) cos 0° = 1


cos 30o =


cos 45o =


cos 60o =


cos 90° = 0


It can be observed that the value of cos θ does not increase in the interval of 0° < θ < 90°


Hence, the given statement is false


(iv) sin θ = cos θ for all values of θ.


This is true when θ = 45°


As,


sin 45o =


cos 45o =


It is not true for all other values of θ


As sin 30o = and cos 30o =


Hence, the given statement is false


(v) cot A is not defined for A = 0°


cot A =


cot 0o =


= =undefined


Hence, the given statement is true


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