Question

State whether the following are true or false. Justify your answer.
(i) $\sin ({\text{A}} + {\text{B)}} = \sin {\text{A}} + \sin {\text{B}}$
(ii) The value of $\sin \theta$ increases as $\theta$ increases.
(iii) The value of $\cos \theta$ increases as $\theta$ increases.
(iv) $\sin \theta = \cos \theta$ for all values of $\theta$ .
(v) $\cot {\text{A}}$ is not defined for ${\text{A}} = 0^\circ$

Answer 1

For the first question we will put the value of some angles into the equation and then we will check whether the given equation is correct or not. For the second question also we will check the values of $\sin \theta$ for some angle and we will decide whether the statement is correct or incorrect. For the third question we will do the same thing which we have done in the second question. For the fourth question also we will check for some values of angles. For the fifth question we will convert $\cot {\text{A}}$ into $\sin {\text{A}}$ and $\cos {\text{A}}$.

Complete step-by-step solution:
(i) Answer: The statement is false. Reason: Let’s take the value ${\text{A}} = 60^\circ$ and ${\text{B}} = 30^\circ$ . Now, put this value of ${\text{A}}$ and ${\text{B}}$ in the equation $\sin ({\text{A}} + {\text{B)}} = \sin {\text{A}} + \sin {\text{B}}$ . Therefore, we get
$\sin (60^\circ + 30^\circ {\text{)}} = \sin 60^\circ + \sin 30^\circ \\\ \Rightarrow \sin \left( {90^\circ } \right) = \dfrac{{\sqrt 3 }}{2} + \dfrac{1}{2} \\\ \Rightarrow 1 = \dfrac{{\sqrt 3 }}{2} + \dfrac{1}{2}$
We know that $1 = \dfrac{{\sqrt 3 }}{2} + \dfrac{1}{2}$ is false.
Hence, we can say that the statement is false.
(ii) Answer: The statement is false. Reason: Let’s check for the value of $\theta = 0,30^\circ ,120^\circ$ . We know that $\sin 0^\circ = 0$, $\sin 30^\circ = \dfrac{1}{2}$ and $\sin 120^\circ = \sin \left( {90^\circ + 30^\circ } \right) = \cos 30^\circ = \dfrac{{\sqrt 3 }}{2}$ . From this example we can say that the value of $\sin \theta$ doesn’t increase as $\theta$ increases.
Hence, the given statement is false.
(iii) Answer: The statement is false. Reason: Let’s check for the value of $\theta = 0,30^\circ ,120^\circ$ . We know that $\cos 0^\circ = 1$ , $\cos 30^\circ = \dfrac{{\sqrt 3 }}{2}$ and $\cos 120^\circ = \cos \left( {90^\circ + 30^\circ } \right) = - \sin 30^\circ = - \dfrac{1}{2}$ . From this example we can say that the value of $\cos \theta$doesn’t increase as $\theta$ increases.
Hence, the given statement is false.
(iv) Answer: The statement is false. Reason: Let’s check for the value of $\theta = 30^\circ$ . We know that $\sin 30^\circ = \dfrac{1}{2}$ and $\cos 30^\circ = \dfrac{{\sqrt 3 }}{2}$ . We got different values of sin and cos for the same value of $\theta$.
Hence, the statement is false.
(v) Answer: The statement is true. Reason: We can write $\cot {\mathbf{A}} = \dfrac{{\cos {\text{A}}}}{{\sin {\text{A}}}} = \dfrac{{\cos 0^\circ }}{{\sin 0^\circ }} = \dfrac{1}{0} = \infty$ and $\infty$ is not defined.
Hence, the given statement is true.

Note: The other important things are the formula of $\sin$ , $\cos$ and $\tan$ which we need to memorize.
$\sin {\text{A = }}\dfrac{{{\text{Opposite}}}}{{{\text{Hypotenuse}}}}$
$\cos {\text{A = }}\dfrac{{{\text{Adjacent}}}}{{{\text{Hypotenuse}}}}$
$\tan {\text{A = }}\dfrac{{{\text{Opposite}}}}{{{\text{Adjacent}}}}$
$\cos ec {\text{A = }}\dfrac{{{\text{Hypotenuse}}}}{{{\text{Opposite}}}}$
$\sec {\text{A = }}\dfrac{{{\text{Hypotenuse}}}}{{{\text{Adjacent}}}}$
$\cot {\text{A = }}\dfrac{{{\text{Adjacent}}}}{{{\text{Opposite}}}}$