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 \(\text{θ}\) increases as \(\text{θ}\) increases.
(iii) The value of cos \(\text{θ}\) increases as \(\text{θ}\) increases.
(iv) sin \(\text{θ}\) = cos \(\text{θ}\) for all values of \(\text{θ}\).
(v) cot A is not defined for A = 0°

Answer 1

(i) Test: sin(A + B) = sin A + sin B. With A = 30° and B = 60°, sin (A + B) = sin (30° + 60°) = sin 90° = 1. Meanwhile, sin A + sin B = sin 30° + sin 60° = \( \frac{1}{2} + \frac{\sqrt{3}}{2} \) = \( \frac{1 + \sqrt{3}}{2} \). Clearly, sin (A + B) ≠ sin A + sin B. Therefore, the statement is false.

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(ii) The value of sin \( \theta \) increases as \( \theta \) increases in the interval \( 0^\circ < \theta < 90^\circ \). Observations: sin 0° = 0, sin 30° = \( \frac{1}{2} \) = 0.5, sin 45° = \( \frac{1}{\sqrt{2}} \) = 0.707, sin 60° = \( \frac{\sqrt{3}}{2} \) = 0.866, sin 90° = 1. Therefore, the statement is true.

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(iii) Values of cos \( \theta \) in the interval \( 0^\circ < \theta < 90^\circ \): cos 0° = 1, cos 30° = \( \frac{\sqrt{3}}{2} \) = 0.866, cos 45° = \( \frac{1}{\sqrt{2}} \) = 0.707, cos 60° = \( \frac{1}{2} \) = 0.5, cos 90° = 0. The value of cos \( \theta \) does not increase in this interval. Therefore, the statement is false.

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(iv) Test: sin \( \theta \) = cos \( \theta \) for all values of \( \theta \). This holds true only when \( \theta = 45^\circ \), as sin 45° = \( \frac{1}{\sqrt{2}} \) and cos 45° = \( \frac{1}{\sqrt{2}} \). It is not true for other values, for example: sin 30° = \( \frac{1}{2} \) while cos 30° = \( \frac{\sqrt{3}}{2} \); sin 60° = \( \frac{\sqrt{3}}{2} \) while cos 60° = \( \frac{1}{2} \); sin 90° = 1 while cos 90° = 0. Therefore, the statement is false.

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(v) Test: cot A is not defined for A = 0°. The formula for cot A is \( \frac{\cos A}{\sin A} \). For A = 0°, cot 0° = \( \frac{\cos 0^\circ}{\sin 0^\circ} = \frac{1}{0} \), which is undefined. Therefore, the statement is true.