Domain and Range of Trigonometric Ratios

Q. Greatest value for sinxcosx

Q.

Prove cot π/24=√2+√3+√4+√6

Q.

If $f\left(x\right)=\cos x\cos 2x\cos 4x\cos 8x\cos 16x$, then $f\left(\frac{\mathrm{\pi }}{4}\right)$ is equal to

1. $\sqrt{2}$

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

3. $0$

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

Q.

How do you find the value of $cot180°$?

Q. In a triangle $ABC$, let $AB=\sqrt{23},$ $BC=3$ and $CA=4$. Then the value of $\frac{\cot A+\cot C}{\cot B}$ is

Q.

Find the degree measures corresponding to the following radian measures:

$\frac{11}{16}$

Q. The number of solutions of the equations $y=\frac{1}{3}[\sin x+[\sin x+\left[\sin x\right]\left]\right]$ and $[y+[y\left]\right]=2\cos x$, where $[.]$ denotes the greatest integer function is

1. $2$
2. $0$
3. $1$
4. infinite

Q.

$Ifcosy=xcos(a+y),withcosa\ne 1,provethat\frac{dy}{dx}=\frac{co{s}^2(a+y)}{sina}.$

Q. $Si{n}^2\left(6x\right)-Si{n}^2\left(4x\right)=Sin2x.Sin10x$

Q.

Set $a,bϵ[-\pi ,\pi ]$ be such that cos (a - b) = 1 and cos (a + b) $=\frac{1}{e}$. The number of pairs of a, b satisfying the above system of equation is

1. 0

2. 1

3. 2

4. 4

Q.

Evaluate $\frac{\sin 9^{\circ }\times \cos 9^{\circ }}{\sin {48}^{\circ }\times \cos {12}^{\circ }}$

Q. The domain of the function $f\left(x\right)=\sec x+\tan x$ is

1. $R-\left\{\right(2n+1)\frac{\pi }{2},n\in Z\}$
2. $R-\{n\pi ,n\in Z\}$
3. $R-\left\{0\right\}$
4. $R$

Q. If $cose{c}^2\theta =\frac{4xy}{(x+y{)}^2}$, then

1. $x=-y$
2. $x=\frac{1}{y}$
3. $x=\frac{1}{x}$
4. $x=y$

Q.

Find the values of the trigonometric function sin 765°

Q.

If $A={\sin }^8\theta +{\cos }^{14}\theta$, then for all real values of $\theta$

1. $A\ge 1$

2. $0<A\le 1$

3. $1<2A\le 3$

4. None of these

Q. If $\cot x=\frac{-5}{12},x$ lies in second quadrant, find the values

Q.

Differentiate the following functions with respect to x.

cos (sin x).

Q.

Let $f:R\to R$ be the function defined by $f\left(x\right)=\frac{1}{2-cos{x}^{\prime }}\forall x\in R$.Then, find the range of f.

Q. Find the number of real roots of the equation $x^2\tan x=1$ between $0\text{and}2\pi$

1. 1
2. 3
3. 2
4. 0

Q.

If $x=e^t\sin t,y=e^t\cos t$, then $\frac{d^{2}y}{d{x}^2}$ at $t=\mathrm{\pi }$ is equal to

1. $2e^{\mathrm{\pi }}$

2. $\frac{1}{2}{e}^{\mathrm{\pi }}$

3. $\frac{1}{2e^{\mathrm{\pi }}}$

4. $\frac{2}{e^{\mathrm{\pi }}}$

Q.

The number of solutions of the equation $\left|\cot \left(x\right)\right|=\cot \left(x\right)+\frac{1}{\sin \left(x\right)}$ in the interval $\left[0,2\pi \right]$ is

Q. Let $f\left(x\right)=\log \left({\log }_{1/3}\right({\log }_7(\sin x+a)\left)\right)$ be defined for every real values of $x$, then the range of $a$

1. $(2,6)$
2. $[2,6]$
3. $(-\infty ,2]\cup (6,\infty )$
4. $(-\infty ,2)\cup [6,\infty )$

Q. In a triangle ABC, if $\sin A\cos B=\frac{1}{4}$ and 3 tan A = tan B, then ${\cot }^{2}A$

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

Q. Number of solutions of the equation $\sin x=\cos x$ in the interval $[-2\pi ,2\pi ]$ is

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

Q. If the equation $x^2+4+3\sin (ax+b)-2x=0$ has at least one real solution, where $a,b\in [0,2\pi ],$ then a possible value of $(a+b)$ can be

1. $\frac{5\pi }{2}$
2. $\frac{7\pi }{2}$
3. $\frac{\pi }{2}$
4. None of these

Q. Consider the equation ${\sin }^{-1}(x^2-6x+\frac{17}{2})+{\cos }^{-1}k=\frac{\pi }{2}.$ Then

1. the largest value of $k$ for which the equation has two distinct real solutions is $1$
2. the equation must have real root if $k\in [-\frac{1}{2},1]$
3. the equation must have real root if $k\in (-1,-\frac{1}{2})$
4. the equation has a unique solution if $k=-\frac{1}{2}$

Q.

If ${\sin }^{-1}\left(\frac{x}{13}\right)+{cosec}^{-1}\left(\frac{13}{12}\right)=\frac{\pi }{2}$ then the value if $x$ is

1. $5$

2. $4$

3. $10$

4. $11$

Q. An angle between the lines whose direction cosines are given by the equations, $l+3m+5n=0$ and $5lm-2mn+6nl=0$, is :

1. ${\cos }^{-1}\left(\frac{1}{3}\right)$
2. ${\cos }^{-1}\left(\frac{1}{6}\right)$
3. ${\cos }^{-1}\left(\frac{1}{4}\right)$
4. ${\cos }^{-1}\left(\frac{1}{8}\right)$

Q. Given that $a,b,c$ are the sides of a triangle $ABC$ which is right angled at $C$, then the minimum value of ${(\frac{c}{a}+\frac{c}{b})}^2$ is

1. $4$
2. $6$
3. $0$
4. $8$

Q. Let $f\left(x\right)={\log }_2(x^2-6x+8).$ Then

1. domain of $f\left(x\right)$ is $R-[2,4]$
2. domain of $f\left(x\right)$ is $(2,4)$
3. range of $f\left(x\right)$ is $R$
4. range of $f\left(x\right)$ is $R^{+}$