Sign of Trigonometric Ratios in Different Quadrants

Q.

If $\text{A + B = 225°}$, then $\left[\frac{cotA}{\mathrm{1\; +}cotA}\right]\left[\frac{cotB}{\mathrm{1\; +}cotB}\right]=$

1. $1$

2. $-1$

3. $0$

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

Q. The value of $\cot \left(\sum _{n=1}^{19}{\cot }^{-1}(1+\sum _{p=1}^{n}2p)\right)$ is :

1. $\frac{22}{23}$
2. $\frac{23}{22}$
3. $\frac{19}{21}$
4. $\frac{21}{19}$

Q.

Find the supplement of $20°+y$.

Q.

If $co{s}^{2}A+co{s}^{2}C=si{n}^{2}B,$ then $△$ ABC is
[MP PET 1991]

1. Right angled

2. Equilateral

3. Isosceles

4. None of these

Q. The value of $\theta$ which satisfy the equation $3{\tan }^2\theta +3\tan \theta -\cot \theta =1$ (where $n\in Z$) can be

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

Q. If $\text{cosec}\theta =\frac{13}{5}$ and $\theta \notin 1^{\text{st}}\text{quadrant}$. Then the value of $\sec \theta$ is

1. $\frac{12}{13}$
2. $-\frac{12}{13}$
3. $\frac{13}{12}$
4. $-\frac{13}{12}$

Q. If $x\in (\pi ,\frac{3\pi }{2})$, then $\sqrt{\frac{1-\sin x}{1+\sin x}}$ is equal to

1. $-\tan x-\sec x$
2. $\sec x-\tan x$
3. $\tan x+\sec x$
4. $\tan x-\sec x$

Q.

Evaluate the value of $cot90°$

Q. The value of $\cot \frac{\pi }{20}\cot \frac{3\pi }{20}\cot \frac{5\pi }{20}\cot \frac{7\pi }{20}\cot \frac{9\pi }{20}$ is

Q.

$x·\left(\frac{dy}{dx}\right)·\tan \left(\frac{y}{x}\right)=y\tan \left(\frac{y}{x}\right)+x$, $y\left(\frac{1}{2}\right)=\frac{\pi }{6}$. Area bounded by $x=0,x=\frac{1}{\sqrt{2}},y=y\left(x\right)$

Q. If $\left[x\right]$ denotes the greatest integer $\le x$, then the system of linear equations $\left[\sin \theta \right]x+[–\cos \theta ]y=0$, $\left[\cot \theta \right]x+y=0$

1. has a unique solution if $\theta \in (\frac{\pi }{2},\frac{2\pi }{3})\cup (\pi ,\frac{7\pi }{6})$
2. have infinitely many solution if $\theta \in (\frac{\pi }{2},\frac{2\pi }{3})\cup (\pi ,\frac{7\pi }{6})$
3. has a unique solution if $\theta \in (\frac{\pi }{2},\frac{2\pi }{3})$ and have infinitely many solutions if $\theta \in (\pi ,\frac{7\pi }{6})$
4. have infinitely many solutions if $\theta \in (\frac{\pi }{2},\frac{2\pi }{3})$ and has a unique solution if $\theta \in (\pi ,\frac{7\pi }{6})$

Q. If $\sin \theta =-\frac{4}{5}$ and $\pi <\theta <\frac{3\pi }{2}$, then $\tan \theta +\cos \theta =$

1. $\frac{11}{15}$
2. $-\frac{11}{15}$
3. $\frac{9}{25}$
4. $-\frac{9}{25}$

Q. $\frac{\tan \left(\frac{35\pi }{6}\right).\sin \left(\frac{11\pi }{3}\right).\sec \left(\frac{7\pi }{3}\right)}{\cot \left(\frac{5\pi }{4}\right).cosec\left(\frac{7\pi }{4}\right).\cos \left(\frac{17\pi }{6}\right)}=$

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

Q. $\mathrm{If}\frac{\sin A}{\sin B}=\frac{\sqrt{3}}{2}\mathrm{and}\frac{\cos A}{\cos B}=\frac{\sqrt{5}}{2}\frac{\pi }{2}<A,B<\pi ,\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\tan A+\tan B\mathrm{is}$

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

Q.

If $cosec2\theta =\sec \left(\theta -33°\right)$, where $2\theta$ is an acute angle, find the value of $\theta$.

Q. $\frac{\tan \left(\frac{35\pi }{6}\right).\sin \left(\frac{11\pi }{3}\right).\sec \left(\frac{7\pi }{3}\right)}{\cot \left(\frac{5\pi }{4}\right).cosec\left(\frac{7\pi }{4}\right).\cos \left(\frac{17\pi }{6}\right)}=$

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

Q. The value of $\sin ({45}^{\circ }+\theta )-\cos ({45}^{\circ }-\theta )$ is

1. $1$
2. $0$
3. $2\cos \theta$
4. $2\sin \theta$

Q. The value of $\frac{1}{2}\tan {585}^{\circ }+\sec (-{660}^{\circ })+\sin {405}^{\circ }\cot {510}^{\circ }-\text{cosec}(-{570}^{\circ })$ is

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

Q. Let $f\left(n\right)=\left(\sin 1\right)\left(\sin 2\right)\left(\sin 3\right)\cdots \left(\sin \right(n\left)\right)\forall n\in N$ where $n$ is in radians. Then the number of elements in the set $A=\left\{f\right(1),f(2),\dots ,f(6\left)\right\}$ that are positive, is

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

Q. $\cos 1^{\circ }+\cos 2^{\circ }+\cos 3^{\circ }+....+\cos {179}^{\circ }$ =

1. \N
2. 1
3. -1
4. 89

Q. The value of ${\text{cot}}^{-1}3+{\text{cosec}}^{-1}\sqrt{5}$ is

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

Q. If $A$ and $B$ are non-zero complementary angles, then the point whose cordinates are $(\sin B+\cos A,\tan B+\cot A)$ lies in

1. First quadrant only
2. Second quadrant only
3. Both first and second quadrant
4. Third quadrant only

Q. If $\cos \theta =-\frac{7}{25}$ and $\pi <\theta <\frac{3\pi }{2}$, then $\tan \theta$ is equal to

1. $-\frac{25}{7}$
2. $\frac{25}{7}$
3. $-\frac{24}{7}$
4. $\frac{24}{7}$

Q.

Find the value of ${\sin }^n{1890}^{\circ }+{\text{cosec}}^n{1890}^{\circ }$. Where $n\in N$

1. $n$

2. $2n$

3. $1$

4. $2$

Q. If $\sqrt{\frac{1-\sin A}{1+\sin A}}+\frac{\sin A}{\cos A}=\frac{1}{\cos A}$ for all permissible values of $A$, then $A$ can belongs to

1. $1^{\text{st}}$ quadrant
2. $2^{\text{nd}}$ quadrant
3. $3^{\text{rd}}$ quadrant
4. $4^{\text{th}}$ quadrant

Q. If the difference between two complementary angles is ${52}^{\circ }$, then the smaller angle is

1. ${20}^{\circ }$
2. ${19}^{\circ }$
3. ${16}^{\circ }$
4. ${17}^{\circ }$

Q.

If $y=|\cos x|+|\sin x|$ then $\frac{dy}{dx}$ at $x=\frac{2\pi }{3}$ is:

1. $\frac{1}{2}(\sqrt{3}+1)$

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

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

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

Q. The value of $\cot 1^{\circ }\cot 2^{\circ }\cot 3^{\circ }\cdots \cot {179}^{\circ }$ is

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

Q. If $x=sin{130}^{\circ }cos{80}^{\circ },y=sin{80}^{\circ }cos{130}^{\circ },z=1+xy,$
which one of the following is true

1. x > 0, y > 0, z > 0
2. x > 0, y < 0, 0 < z < 1
3. x > 0, y < 0, z > 1
4. x < 0, y < 0, 0 < z < 1

Q. $\mathrm{if}\sin \theta =0\Rightarrow \theta =\mathrm{n\pi },\mathrm{where}n\in Z$

1. False
2. True