Trigonometric Ratios Using Right Angled Triangle

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.

A tower stands at the center of a circular park.

$A$ and $B$ are two points on the boundary of the park such that $(AB=a)$subtends an angle of $60°$ at the foot of the tower and the angle of elevation of the top of the first tower from $A$ or $B$ is $30°$.

The height of the tower is

1. $\frac{2a}{\sqrt{3}}$

2. $2a\sqrt{3}$

3. $\frac{a}{\sqrt{3}}$

4. $\sqrt{3}$

Q.

The value of $\text{θ}$ satisfying $\text{sin7θ=sin4θ-sinθ}$ and $0<\text{θ<}\frac{\mathrm{\pi }}{2}$ are

Q.

An aeroplane flying at a height of $9000\text{m}$ vertically above from the ground, passes another aeroplane when the angles of elevation of the two aeroplanes from a point on the ground are ${60}^{\circ }$ and ${30}^{\circ }$ respectively. Then the vertical distance between the aeroplanes at that instant is

1. $1000\text{m}$
2. $6000\text{m}$
3. $8000\text{m}$
4. $3000\text{m}$

Q. A tower $T_1$ of height $60$ m is located exactly opposite to a tower $T_2$ of height $80m$ on a straight road. From the top of $T_1$, if the angle of depression of the foot of $T_2$ is twice the angle of elevation of the top of $T_2$, then the width (in $m$) of the road between the feet of the towers $T_1$ and $T_2$ is:

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

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.

Let $A_0{A}_1{A}_2{A}_3{A}_4{A}_5$ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments $A_0{A}_1,A_0{A}_{2}and{A}_0{A}_4$ is

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

2. $3\sqrt{3}$

3. $3$

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

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

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

Q. The angle of elevation of the top of a pole from the point $A$ on ground is ${45}^{\circ },$ whereas the angle of depression of the foot of pole from the point which is $10$ feet above point $A$ is ${30}^{\circ }.$ Then the length of the pole is

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

Q.

prove that 1 + sec 20 = cot 40 cot 30

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

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. 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. Which among the following represents the graph of curve $y=2\sin 2x$

1. 
2. 
3. 
4. 

Q. The solution set of $cos2\theta =co{s}^2\theta$ is

1. 
2. 
3. 
4. 

Q. The angle of elevation of a cloud $C$ from a point $P,200$ m above a still lake is ${30}^{\circ }$. If the angle of depression of the image of $C$ in the lake from the point $P$ is ${60}^{\circ }$, then $PC$ (in m) is equal to

1. $200\sqrt{3}$
2. $400\sqrt{3}$
3. $400$
4. $100$

Q. More than One Answer Type
एक से अधिक उत्तर प्रकार के प्रश्न

If tan²α + 2 tanα × tan2β = tan²β + 2 tanβ × tan2α, then which of the following can be true?

यदि tan²α + 2 tanα × tan2β = tan²β + 2 tanβ × tan2α, तब निम्न में से कौनसा विकल्प सही हो सकता है?

1. tan²α + 2 tanα × tan2β = 0
2. tanα + tanβ = 0
3. tan²β + 2 tanβ × tan2α = 1
4. tanα = tanβ

Q. If $y=\sin \left(\sin x\right)$, then the value of $y^{\prime \prime }+\left(\tan x\right)y^{\prime }+y{\cos }^{2}x+5$ is

Q. In any triangle ABC, prove the following:

$a\sin \frac{A}{2}\sin \left(\frac{B-C}{2}\right)+b\sin \frac{B}{2}\sin \left(\frac{C-A}{2}\right)+c\sin \frac{C}{2}\sin \left(\frac{A-B}{2}\right)=0$

Q.

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

1. 3

2. 9

3. 

4. None of these

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

Q. In $△ABC,$ sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively. If $b=c\sin B,$ then the triangle is

1. Equilateral
2. Right angled
3. Acute angle
4. Obtuse angle

Q.

prove that tan 70 -tan50 +tan10 =tan60

Q. The angle of elevation of a jet plane from a point $A$ on the ground is ${60}^{\circ }.$ After a flight of $20$ seconds at the speed of $432\text{km/hour},$ the angle of elevation changes to ${30}^{\circ }.$ If the jet plane is flying at a constant height, then its height is

1. $1200\sqrt{3}$ m
2. $1800\sqrt{3}$ m
3. $3600\sqrt{3}$ m
4. $2400\sqrt{3}$ m

Q.

$\frac{si{n}^{3}A+co{s}^{3}A}{sinA+cosA}+\frac{si{n}^{3}A-co{s}^{3}A}{sinA-cosA}=2$

Q. If $0\le \alpha ,\beta \le {90}^{\circ }$ and $tan(\alpha +\beta )=3$ and $tan(\alpha -\beta )=2$, then the value of $sin2\alpha$ is -

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

Q. The value of $\left(4{\oplus }_{5}2\right){\oplus }_{5}3$ is

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