Circular Measurement of Angle

Q.

The greatest and least value of $\sin x\cos x$are

1. $1,-1$

2. $\frac{1}{2},-\frac{1}{2}$

3. $\frac{1}{4},-\frac{1}{4}$

4. $2,-2$

Q. If $\sin x=\frac{1}{3}$ and $\cos x<0$, then the value of $\tan 3x$ is

1. $\frac{23}{10\sqrt{2}}$
2. $-\frac{23}{10\sqrt{2}}$
3. $-\frac{23}{5\sqrt{2}}$
4. $-\frac{23}{15\sqrt{2}}$

Q. The value of $\frac{\cos {58}^{\circ }}{\sin {32}^{\circ }}+\frac{\sin {22}^{\circ }}{\cos {68}^{\circ }}-\frac{\cos {38}^{\circ }\text{cosec}{52}^{\circ }}{\tan {18}^{\circ }\tan {35}^{\circ }\tan {72}^{\circ }\tan {55}^{\circ }}$
is

1. $0$
2. $\frac{1}{2}$
3. $1$
4. $2$

Q. If $\sin x=\frac{12}{13}$ and $x\in [0,\pi ]$, then the possible value(s) of $\sec x+\tan x$ is/are

1. $5$
2. $4$
3. $-5$
4. $-4$

Q.

$x^o=\frac{\pi }{180}.x$ radian and $xradian=\frac{180}{\pi }\times x$ degree

1. True

2. False

Q.

How many of the following are matched correctly?

$\begin{array}{cc}\text{Degree measurement}& \text{Radian measurement}\\ \text{(A)}{180}^{\circ }& \text{(1)}\pi \\ \text{(B)}{60}^{\circ }& \text{(2)}\frac{\pi }{6}\\ \text{(C)}{0}^{\circ }& \text{(3)}0\\ \text{(D)}{120}^{\circ }& \text{(4)}2\frac{\pi }{6}\\ \text{(E)}{360}^{\circ }& \text{(5)}2\pi \\ \text{(F)}{30}^{\circ }& \text{(6)}\frac{\pi }{3}\\ \text{(G)}{90}^{\circ }& \text{(7)}\frac{\pi }{2}\\ \text{(H)}{45}^{\circ }& \text{(8)}\frac{\pi }{4}\\ \text{(I)}{270}^{\circ }& \text{(9)}3\pi \end{array}$
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Q. If both $\theta$ and $\varphi$ are acute angles such that $\sin \theta =\frac{1}{2}$ and $\cos \varphi =\frac{1}{3}$, then $(\theta +\varphi )$ belongs to

1. $(\frac{\pi }{3},\frac{5\pi }{12})$
2. $(\frac{\pi }{2},\frac{2\pi }{3})$
3. $(\frac{2\pi }{3},\frac{5\pi }{6})$
4. $(\frac{5\pi }{12},\frac{\pi }{2})$

Q. If $\sin x=\frac{12}{13}$ and $x\in [0,\pi ]$, then the possible value(s) of $\sec x+\tan x$ is/are

1. $5$
2. $4$
3. $-5$
4. $-4$

Q. The value of ${\log }_{\text{cosec}\pi /4}|\cos 2019\pi |$ is

1. $1$
2. $-1$
3. $0$
4. $\sqrt{2}$

Q. If in a right angle $△ABC$, acute angles $A$ and $B$ satisfy $\tan A+\tan B+{\tan }^{2}A+{\tan }^{2}B=10$, then which of the following is/are correct?

1. $\tan A=\frac{3+\sqrt{5}}{2}$
2. $\tan A=\frac{3-\sqrt{5}}{2}$
3. $\tan A=-2-\sqrt{3}$
4. $\tan A=-2+\sqrt{3}$

Q. If $A$ and $B$ are complemetary angles, then $\sqrt{\frac{\cos A}{\sin B}}-\cos A\sin B=$

1. $\cos A\sin A$
2. ${\cos }^{2}A$
3. ${\sin }^{2}A$
4. ${\sin }^{2}B$

Q. If in a right angle $△ABC$, acute angles $A$ and $B$ satisfy $\tan A+\tan B+{\tan }^{2}A+{\tan }^{2}B=10$, then which of the following is/are correct?

1. $\tan A=\frac{3+\sqrt{5}}{2}$
2. $\tan A=\frac{3-\sqrt{5}}{2}$
3. $\tan A=-2-\sqrt{3}$
4. $\tan A=-2+\sqrt{3}$