State whether the following are true or false. Justify your answer. (i) sin (A + B) = sin A + sin B.

Question

(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°.

Answer 1

(i) It is not true, in general. This relation is true for some particular values of A and B.

For example, if A = 0 and B = 90,

Then, Sin (A+B) = sin (0+90) = sin 90 = 1

And, sin A + sin B = sin 0 + sin 90 = 0 + 1 = 1

Hence, sin (A+B) = sin A + sin B

(ii) True.

sin 0° = 0

sin 30° = 1/2

sin 45° = 1/√2

sin 60° = √3/2

sin  90° = 1

Thus the value of sin θ increases as θ increases.

(iii) False.

cos 0° = 1

cos 30° = √3/2

cos 45° = 1/√2

cos 60° = 1/2

cos 90° = 0

Thus the value of cos θ decreases as θ increases.

(iv) True.

cot A = cos A/sin A

cot 0° = cos 0°/sin 0° = 1/0 = undefined