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°.

Question

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°.

KnowledgeBoat · Accepted Answer

(i) Let A = 30° and B = 60°. Substituting value of A and B in L.H.S. of the equation sin (A + B) = sin A + sin B, we get : ⇒ sin(A + B) ⇒ sin(30° + 60°) ⇒ sin 90° ⇒ 1. Substituting value of A and B in R.H.S. of the equation sin (A + B) = sin A + sin B, we get : ⇒ sin A + sin B ⇒ sin 30° + sin 60° ⇒ ⇒ . Since, sin (A + B) sin A + sin B. Hence, statement sin (A + B) = sin A + sin B is false. (ii) Let θ be 0°, 30° and 45°. sin 0° = 0, sin 30° = and sin 45° = . We see that, sin θ increases as θ increases. Hence, the statement the value of sin θ increases as θ increases is true. (iii) Let θ be 0°, 30° and 45°. cos 0° = 1, cos 30° = and cos 45° = . We see that, cos θ decreases as θ increases. Hence, the statement the value of cos θ increases as θ increases is false. (iv) Let θ be 0°, 30° and 45°. sin 0° = 0, sin 30° = and sin 45° = , cos 0° = 1, cos 30° = and sin 45° = , From above we see that, sin θ = cos θ, only when θ = 45°. Hence, the statement sin θ = cos θ for all values of θ is false. (v) We know that, cot A = For A = 0°, sin A = 0. ∴ cot A is not defined. Hence, the statement cot A is not defined for A = 0° is true.

Mathematics

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°.

Trigonometric Identities

Related Questions

Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

Write all the other trigonometric ratios of ∠A in terms of sec A.

Mathematics

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°.

Trigonometric Identities

Related Questions

Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

Write all the other trigonometric ratios of ∠A in terms of sec A.

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°.