Trigonometric Ratios Using Right Angled Triangle

Q. If $\sec \theta =max(x+\frac{1}{x}),x\in R,$ where $x<0$, then the value of $\theta$

1. ${60}^{\circ }$
2. ${30}^{\circ }$
3. ${120}^{\circ }$
4. ${150}^{\circ }$

Q. Two poles standing on a horizontal ground are of heights $5$ m and $10$ m respectively. The line joining their tops makes an angle of ${15}^{\circ }$ with ground. Then the distance (in m) between the poles, is :

1. $\frac{5}{2}(2+\sqrt{3})$
2. $10(\sqrt{3}-1)$
3. $5(2+\sqrt{3})$
4. $5(\sqrt{3}+1)$

Q.

The greatest and least values of $(si{n}^{-1}x{)}^3+(co{s}^{-1}x{)}^3$ are

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

2. $\frac{-{\pi }^3}{2},\frac{{\pi }^3}{2}$

3. $\frac{{\pi }^3}{32},\frac{7{\pi }^3}{8}$

4. None

Q. An aeroplane flying at a constant speed, parallel to the horizontal ground, $\sqrt{3}km$ above it, is observed at an elevation of ${60}^{\circ }$ from a point on the ground. If, after five seconds, its elevation from the same point, is ${30}^{\circ }$, then the speed (in $km/hr$) of the aeroplane, is :

1. $1500$
2. $1440$
3. $750$
4. $720$

Q. If $f\left(x\right)=({\sin }^{-1}x{)}^2+({\cos }^{-1}x{)}^2$, then

1. $f\left(x\right)$ has the least value of $\frac{{\pi }^2}{8}$
2. $f\left(x\right)$ has the greatest value of $\frac{5{\pi }^2}{8}$
3. $f\left(x\right)$ has the least value of $\frac{{\pi }^2}{16}$
4. $f\left(x\right)$ has the greatest value of $\frac{5{\pi }^2}{4}$

Q. Three poles $A,B$ and $C$ are in a straight line, apart by $10$ metres each. The height of pole $A$ is $20$ metres and the angle of depression from the top of pole $A$ to the top of pole $B$ is ${60}^{\circ }$ and the angle of elevation from the top of pole $B$ to the top of pole $C$ is ${30}^{\circ }.$ The height (in metres) of pole $C$ is

1. $\frac{10}{3}(3-\sqrt{3})$
2. $\frac{20}{3}(3+\sqrt{3})$
3. $\frac{20}{3}(3-\sqrt{3})$
4. $\frac{10}{3}(3+\sqrt{3})$

Q.

From given figure, $\sin x+\sec x+\tan x$ is $\alpha$, find $156\alpha$.

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Q. Two posts are $20$ metre apart and the height of one post is double that of the other. From the mid-point of the line segment joining their feet, an observer finds that the angular elevation of their tops are complementary. Then the height of the shorter post (in metre) is

1. $5\sqrt{2}$
2. $5$
3. $20\sqrt{3}$
4. $10$

Q. If $\sec \theta =min(a^x+\frac{1}{a^x}),$ where $a>0,x\in R,$ then the value of $\theta$

1. ${120}^{\circ }$
2. ${30}^{\circ }$
3. ${60}^{\circ }$
4. ${150}^{\circ }$

Q. If $a,b,c$ are the sides of a triangle $ABC$ opposite to angles $A,B,C$ respectively and angle $C$ is ${90}^{\circ },$ then $\tan A+\tan B$ is equal to

1. $\frac{a^2}{bc}$
2. $\frac{b^2}{ac}$
3. $\frac{c^2}{ab}$
4. $\frac{c}{ab}$

Q.

Match the following:

Given $\sin x$ = $\left(\frac{2}{5}\right)$ and $x\in (0,\frac{\pi }{2})$

$\begin{array}{cc}\text{(p)}\cos x& \text{(1)}\frac{5}{2}\\ \text{(q)}\tan x& \text{(2)}\frac{\sqrt{21}}{2}\\ \text{(r)}\csc x& \text{(3)}\frac{\sqrt{21}}{5}\\ & \text{(4)}\frac{2}{\sqrt{21}}\end{array}$

1. P-2, Q-3, R-4

2. P-3, Q-2, R-4

3. P-2, Q-3, R-1

4. None of these

Q. In any $\Delta ABC$, the least value of $\left(\frac{si{n}^{2}A+sinA+1}{sinA}\right)$ is

1. 3
2. $\sqrt{3}$
3. 9
4. None of these

Q. If $2\sec \theta \text{cosec}\theta -\cot \theta =3$, then the value of $\tan \theta$ is/are

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

Q. A bird is sitting on the top of a vertical pole $20m$ high and its elevation from a point $O$ on the ground is ${45}^{\circ }$. It flies off horizontally straight away from the point $O$. After one second, the elevation of the bird from $O$ is reduced to ${30}^{\circ }$. Then the speed (in $m/s$) of the bird is

1. $40(\sqrt{2}-1)$
2. $40(\sqrt{3}-\sqrt{2})$
3. $20\sqrt{2}$
4. $20(\sqrt{3}-1)$

Q. The value of $(\cos \theta +\sin \theta +1)(\cos \theta +\sin \theta -1)-2\sin \theta \cos \theta$ is

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

Q. If $x=r\sin A\sin B,y=r\cos A\sin B,z=r\cos B,$ then $x^2+y^2+z^2$ is equal to

1. $2r^2$
2. $r^2$
3. $3r^2$
4. $r^2+1$

Q. The top of a hill observed from the top and bottom of a building of height ${}^{\prime }{h}^{\prime }$ is at angles of elevation $p$ and $q$, respectively. The height of hill

1. $\frac{h\cot q}{\cot q-\cot p}$
2. $\frac{h\cot p}{\cot p-\cot q}$
3. $\frac{h\tan p}{\tan p-\tan q}$
4. None of these