Allied Angles

Q. The value of $2\sin \left(\frac{\pi }{8}\right)\sin \left(\frac{2\pi }{8}\right)\sin \left(\frac{3\pi }{8}\right)\sin \left(\frac{5\pi }{8}\right)\sin \left(\frac{6\pi }{8}\right)\sin \left(\frac{7\pi }{8}\right)$ is

1. $\frac{1}{8\sqrt{2}}$
2. $\frac{1}{4\sqrt{2}}$
3. $\frac{1}{8}$
4. $\frac{1}{4}$

Q. Which of the following is the principal value of ${\text{cosec}}^{-1}x$

1. $(-\frac{\pi }{2},\frac{\pi }{2})$
2. $(0,\pi )-\left\{\frac{\pi }{2}\right\}$
3. $[-\frac{\pi }{2},\frac{\pi }{2}]$
4. $[-\frac{\pi }{2},\frac{\pi }{2}]-\left\{0\right\}$

Q. The principal value of ${\text{cosec}}^{-1}(-2)$ is

[1 mark]

1. $-\frac{\pi }{6}$
2. $\frac{5\pi }{6}$
3. $-\frac{\pi }{3}$
4. $\frac{2\pi }{3}$

Q.

If $A+B+C=\pi$, then $\sin 2A+\sin 2B+\sin 2C$ is equal to

1. $4\sin A\sin B\sin C$

2. $4\cos A\cos B\cos C$

3. $2\cos A\cos B\cos C$

4. $2\sin A\sin B\sin C$

Q. The value of $\frac{{(sin\frac{\pi }{8}+icos\frac{\pi }{8})}^8}{{(sin\frac{\pi }{8}-icos\frac{\pi }{8})}^8}$ is

1. -1
2. 0
3. 1
4. 2i

Q. The value of ${\sin }^2\frac{\pi }{4}+{\sin }^2\frac{3\pi }{4}+{\sin }^2\frac{5\pi }{4}+{\sin }^2\frac{7\pi }{4}$ is

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

Q. Find the degree measure of the angle subtended at the center of a circle of radius $100$ cm by an arc of length $22$ cm.
$(\text{use}\pi =\frac{22}{7})$

Q. The value of $\cot A+\tan (180°+A)+\tan (90°+A)+\tan (360°-A)$ is

Q. The value of ${\cos }^{-1}\left(\cos {680}^{\circ }\right)$ is

1. $\frac{2\pi }{9}$
2. $\frac{-2\pi }{9}$
3. $\frac{34\pi }{9}$
4. $\frac{\pi }{9}$

Q. If $\cos A=n\cos B$ and $\sin A=m\sin B$, then $(m^2-n^2){\sin }^{2}B=$

1. $1$
2. $1+n^2$
3. $n^2$
4. $1-n^2$

Q.

The sum of the series $\sum _{\mathrm{n}=1}^{\infty }\sin \left(\frac{\mathrm{n}!\mathrm{\pi }}{720}\right)$ is

1. $\sin \left(\frac{\mathrm{\pi }}{180}\right)+\sin \left(\frac{\mathrm{\pi }}{360}\right)+\sin \left(\frac{\mathrm{\pi }}{540}\right)$

2. $\sin \left(\frac{\mathrm{\pi }}{6}\right)+\sin \left(\frac{\mathrm{\pi }}{30}\right)+\sin \left(\frac{\mathrm{\pi }}{120}\right)+\sin \left(\frac{\mathrm{\pi }}{360}\right)$

3. $\sin \left(\frac{\mathrm{\pi }}{6}\right)+\sin \left(\frac{\mathrm{\pi }}{30}\right)+\sin \left(\frac{\mathrm{\pi }}{120}\right)+\sin \left(\frac{\mathrm{\pi }}{360}\right)+\sin \left(\frac{\mathrm{\pi }}{720}\right)$

4. $\sin \left(\frac{\mathrm{\pi }}{180}\right)+\sin \left(\frac{\mathrm{\pi }}{360}\right)$

Q.

If the arcs of the same length in two circles substend angles ${65}^{\circ }and{110}^{\circ }$ at the centre, find the ratio of their radii.

Q. The angle subtended by an arc of length $2\pi r$ of a circle with radius $r$ equals $2\pi$ radians.

1. False
2. True

Q. If $\theta =178°$, then the value of $\sin \theta \sqrt{1+{\cot }^2\theta }+\cos \theta \sqrt{1+{\tan }^2\theta }$ is

1. $2$
2. $-2$
3. $0$
4. $1$

Q. The value of ${\sin }^{-1}\left(\sin \frac{5\pi }{6}\right)$ is

1. $\frac{\pi }{6}$
2. $\frac{5\pi }{6}$
3. $\frac{7\pi }{6}$
4. $-\frac{\pi }{6}$

Q. ${\sin }^{-1}(1-x)-2{\sin }^{-1}x=\frac{\pi }{2},$ then $x$ is equal to

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

Q. If in two circles, arcs of the same length subtend angles ${60}^{\circ }$ and ${75}^{\circ }$ at the centre, find the ratio of their radii.

Q.

The line OA makes an angle of $\theta ={30}^o$ with negative x-axis as shown in the figure. Which of the following could be the measure of the angle made by OA with respect to positive x-axis?

1. $-{30}^o$

2. $-{150}^o$

3. ${210}^o$

4. ${150}^o$

Q. The arcs of the same length in two circles subtend angles of ${28}^{\circ }$ and ${35}^{\circ }$ at their centres. Then the ratio of their respective radii is

1. $3:4$
2. $4:3$
3. $4:5$
4. $5:4$

Q. The value of ${\left(\frac{1+\sin \frac{2\pi }{9}+i\cos \frac{2\pi }{9}}{1+\sin \frac{2\pi }{9}-i\cos \frac{2\pi }{9}}\right)}^3$ is:

1. $-\frac{1}{2}(1-i\sqrt{3})$
2. $\frac{1}{2}(1-i\sqrt{3})$
3. $-\frac{1}{2}(\sqrt{3}-i)$
4. $\frac{1}{2}(\sqrt{3}-i)$

Q. For the given table choose the correct option
$\begin{array}{cccc}& & & \\ & \text{Column I}& & \text{Column II}\\ \left(a\right)& \text{The value of}\cot \left(\frac{41\pi }{4}\right)\text{is}& \left(p\right)& 1\\ \left(b\right)& \text{The value of}\sec (-{600}^{\circ })\text{is}& \left(q\right)& 2\\ \left(c\right)& \text{The value of}{\text{cosec}}^2\left(\frac{41\pi }{4}\right)\text{is}& \left(r\right)& -2\\ \left(d\right)& \text{The value of}\tan \left(\frac{19\pi }{4}\right)\text{is}& \left(s\right)& -1\end{array}$

1. $\left(a\right)\to \left(p\right),\left(b\right)\to \left(r\right),\left(c\right)\to \left(q\right),\left(d\right)\to \left(s\right)$
2. $\left(a\right)\to \left(p\right),\left(b\right)\to \left(q\right),\left(c\right)\to \left(r\right),\left(d\right)\to \left(s\right)$
3. $\left(a\right)\to \left(q\right),\left(b\right)\to \left(r\right),\left(c\right)\to \left(p\right),\left(d\right)\to \left(s\right)$
4. $\left(a\right)\to \left(s\right),\left(b\right)\to \left(r\right),\left(c\right)\to \left(q\right),\left(d\right)\to \left(p\right)$

Q. Write the inclination of a line which is Perpendicular to y-axis

Q.
In any $\Delta ABC,\sum \left(\frac{{\sin }^{2}A+\sin A+1}{\sin A}\right)$ is always greater than

1. 9
2. 3
3. 27
4. $\frac{1}{3}$

Q. Show that $(cosec\Theta -cot\Theta {)}^2=\frac{1-cos\Theta }{1+cos\Theta }$

Q. For a circle of radius $r=2m$, the angle subtended by an arc of length $\frac{\pi }{2}m$ will be

1. 
   $\pi \mathrm{radians}$
2. 
   $\frac{\pi }{2}\mathrm{radians}$
3. 
   $\frac{\pi }{8}\mathrm{radians}$
4. 
   $\frac{\pi }{4}\mathrm{radians}$

Q. $\frac{\sin x}{a}=\frac{\cos x}{b}$, prove that a $\sin 2x+b\cos 2x=b.$

Q. The value of $\cot A+\tan (180°+A)+\tan (90°+A)+\tan (360°-A)$ is

Q. $\sin x+{\sin }^{2}x=1,$ then the value of ${\cos }^{12}x+3{\cos }^{10}x+3{\cos }^{8}x+{\cos }^{6}x-1$ is

1. $-1$
2. $0$
3. $1$
4. $2$

Q. Prove that
$\cot A+\tan ({180}^{\circ }+A)+\tan ({90}^{\circ }+A)+\tan ({360}^{\circ }-A)=0$.

Q. The value of ${\sin }^{-1}\left(\sin \frac{2\pi }{3}\right)$ is

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