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R.D. Sharma solutions 10 Mathematics chapter 5 Trigonometric Ratios

Chapters

R.D. Sharma 10 Mathematics

10 Mathematics

Chapter 5 - Trigonometric Ratios

Page 0

Prove the following trigonometric identities:

`(1 - cos^2 A) cosec^2 A = 1`

A tower stands vertically on the ground. From a point on the ground 20 m away from the foot of the tower, the angle of elevation of the top of the tower is 60°. What is the height of the tower?

In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`sin A = 2/3`

In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`cos A = 4/5`

In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.

tan θ = 11

In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`sin theta = 11/5`

In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.

`tan alpha = 5/12`

In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`sin theta = sqrt3/2`

In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`cos theta = 7/25`

In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.

`tan theta = 8/15`

In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.

`cot theta = 12/5`

In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`sec theta = 13/5`

In the following, trigonometric ratios are given. Find the values of the other trigonometric ratios.

`cosec theta = sqrt10`

In the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.

`cos theta = 12/2`

Prove the following trigonometric identities

(1 + cot2 A) sin2 A = 1

A tower stands vertically on the ground. From a point on the ground, 20 m away from the foot of the tower, the angle of elevation of the top of the tower is 600. What is the height of the tower?

In a ΔABC, right angled at B, AB = 24 cm, BC = 7 cm. Determine

Sin A, Cos A

In a ΔABC, right angled at B, AB = 24 cm, BC = 7 cm. Determine

Sin C, cos C

A ladder is placed along a wall of a house such that its upper end is touching the top of the wall. The foot of the ladder is 2 m away from the wall and the ladder is making an angle of 60° with the level of the ground. Determine the height of the wall.

Prove the following trigonometric identities.

tan2θ cos2θ = 1 − cos2θ

In Fig below, Find tan P and cot R. Is tan P = cot R?

If `sin A = 9/41` compute cos 𝐴 𝑎𝑛𝑑 tan 𝐴

Prove the following trigonometric identities.

`cosec theta sqrt(1 - cos^2 theta) = 1`

An electric pole is 10 m high. A steel wire tied to the top of the pole is affixed at a point on the ground to keep the pole upright. If the wire makes an angle of 45° with the horizontal through the foot of the pole, find the length of the wire.

A kit is flying at a height of 75 metres from the ground level, attached to a string inclined at 60 to the horizontal. Find the length of the string to the nearest metre.

Prove the following trigonometric identities.

(sec2 θ − 1) (cosec2 θ − 1) = 1

Given 15 cot A = 8. Find sin A and sec A

The length of a string between a kite and a point on the ground is 90 meters. If the string makes an angle O with the ground level such that tan O = 15/8, how high is the kite? Assume that there is no slack in the string.

Prove the following trigonometric identities.

`tan theta + 1/tan theta = sec theta cosec theta`

In ΔPQR, right angled at Q, PQ = 4 cm and RQ = 3 cm. Find the values of sin P, sin R, sec P and sec R.

if `cot theta = 1/sqrt3` find the value of `(1 - cos^2 theta)/(2 - sin^2 theta)`

Prove the following trigonometric identities

`cos theta/(1 - sin theta) = (1 + sin theta)/cos theta`

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff. At a point on the plane 70 metres away from the tower, an observer notices that the angles of elevation of the top and the bottom of the flagstaff are respectively 60° and 45°. Find the height of the flag-staff and that of the tower.

If cot θ = 7/8 evaluate `((1+sin θ )(1-sin θ))/((1+cos θ)(1-cos θ))`

If cot θ = 7/8, evaluate cot2 θ

A vertically straight tree, 15 m high, is broken by the wind in such a way that its top just touches the ground and makes an angle of 60° with the ground. At what height from the ground did the tree break?

If 3 cot A = 4, Check whether `((1-tan^2 A)/(1+tan^2 A)) = cos^2 A - sin^2 A` or not

Prove the following trigonometric identities.

`cos theta/(1 + sin theta) = (1 - sin theta)/cos theta`

If `tan theta = a/b`, find the value of `(cos theta + sin theta)/(cos theta - sin theta)`

Prove the following trigonometric identities

`cos^2 A + 1/(1 + cos^2 A) = 1`

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag-staff of height 5 meters. At a point on the plane, the angles of elevation of the bottom and the top of the flag-staff are respectively 300 and 600. Find the height of the tower.

If 3 tan θ = 4, find the value of `(4cos theta - sin theta)/(2cos theta + sin theta)`

A person observed the angle of elevation of the top of a tower as 30°. He walked 50 m towards the foot of the tower along level ground and found the angle of elevation of the top of the tower as 60°. Find the height of the tower.

Prove the following trigonometric identities.

`sin^2 A + 1/(1 + tan^2 A) = 1`

If 3 cot θ = 2, find the value of `(4sin theta - 3 cos theta)/(2 sin theta + 6sin theta)`

The shadow of a tower, when the angle of elevation of the sun is 45°, is found to be 10 m. longer than when it was 600. Find the height of the tower.

Prove the following trigonometric identities.

`sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta`

If tan θ = `a/b` prove that `(a sin theta - b cos theta)/(a sin theta + b cos theta) = (a^2 - b^2)/(a^2 + b^2)`

Prove the following trigonometric identities.

`(1 - cos theta)/sin theta = sin theta/(1 + cos theta)`

A parachutist is descending vertically and makes angles of elevation of 45° and 60° at two observing points 100 m apart from each other on the left side of himself. Find the maximum height from which he falls and the distance of the point where he falls on the ground form the just observation point.

Prove the following trigonometric identities.

`sin theta/(1 - cos theta) =  cosec theta + cot theta`

if sec theta = 13/5 show that `(2 cos theta - 3 cos theta)/(4 sin theta - 9 cos theta) = 3`

On the same side of a tower, two objects are located. When observed from the top of the tower, their angles of depression are 45° and 60°. If the height of the tower is 150 m, find the distance between the objects.

Prove the following trigonometric identities.

`(1 - sin theta)/(1 + sin theta) = (sec theta - tan theta)^2`

The angle of elevation of a tower from a point on the same level as the foot of the tower is 30°. On advancing 150 metres towards the foot of the tower, the angle of elevation of the tower becomes 60°.  Show that the height of the tower is 129.9 metres (Use `sqrt3 = 1.732`)

If `cos theta = 12/13`, show that `sin theta (1 - tan theta) = 35/156`

Prove the following trigonometric identities.

(cosecθ + sinθ) (cosecθ − sinθ) = cot2 θ + cos2θ

If `cot theta = 1/sqrt3` show that  `(1 - cos^2 theta)/(2 - sin^2  theta) = 3/5`

The angle of elevation of the top of a tower as observed form a point in a horizontal plane through the foot of the tower is 32°. When the observer moves towards the tower a distance of 100 m, he finds the angle of elevation of the top to be 63°. Find the height of the tower and the distance of the first position from the tower. [Take tan 32° = 0.6248 and tan 63° = 1.9626]

If `tan theta = 1/sqrt7`     `(cosec^2 theta - sec^2 theta)/(cosec^2 theta + sec^2 theta) = 3/4`

The angle of elevation of the top of a tower from a point A on the ground is 30°. Moving a distance of 20metres towards the foot of the tower to a point B the angle of elevation increases to 60°. Find the height of the tower & the distance of the tower from the point A.

Prove the following trigonometric identities.

`((1 + cot^2 theta) tan theta)/sec^2 theta = cot theta`

if `sin theta = 12/13` find `(sin^2 theta - cos^2 theta)/(2sin theta cos theta) xx 1/(tan^2 theta)`

Prove the following trigonometric identities.

(secθ + cosθ) (secθ − cosθ) = tan2θ + sin2θ

From the top of a building, 15 m high the angle of elevation of the top of a tower is found to be 30°. From the bottom of the same building, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower and the distance between the tower and building.

Prove the following trigonometric identities.

secA (1 − sinA) (secA + tanA) = 1

if `sec theta = 5/4` find the value of `(sin theta - 2 cos theta)/(tan theta - cot theta)`

On a horizontal plane, there is a vertical tower with a flagpole on the top of the tower. At a point 9 meters away from the foot of the tower the angle of elevation of the top and bottom of the flagpole are 60° and 30° respectively. Find the height of the tower and the flagpole mounted on it.

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30 ° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

if `cos theta = 5/13` find the value of `(sin^2 theta - cos^2 theta)/(2 sin theta cos theta) = 3/5`

Prove the following trigonometric identities.

(cosecA − sinA) (secA − cosA) (tanA + cotA) = 1

if `tan theta = 12/13` Find `(2 sin theta cos theta)/(cos^2 theta - sin^2 theta)`

Prove the following trigonometric identities.

`tan^2 theta - sin^2 theta tan^2 theta sin^2 theta`

From a point P on the ground the angle of elevation of a 10 m tall building is 30°. A flag is hoisted at the top of the building and the angle of elevation of the top of the flag-staff from P is 45°. Find the length of the flag-staff and the distance of the building from the point P. (Take `sqrt3` = 1.732)

if `cos theta = 3/5`, find the value of `(sin theta - 1/(tan theta))/(2 tan theta)`

Prove the following trigonometric identities.

(1 + tan2θ) (1 − sinθ) (1 + sinθ) = 1

A 1.6 m tall girl stands at a distance of 3.2 m from a lamp-post and casts a shadow of 4.8 m on the ground. Find the height of the lamp-post by using (i) trigonometric ratios (ii) property of similar triangles.

A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building

if `sin theta = 3/5  " evaluate " (cos theta - 1/(tan theta))/(2 cot theta)`

Prove the following trigonometric identities.

sin2 A cot2 A + cos2 A tan2 A = 1

The shadow of a tower standing on a level ground is found to be 40 m longer when Sun’s altitude is 30° than when it was 60°. Find the height of the tower

 if `sec A = 5/4` verify that `(3 sin A - 4 sin^3 A)/(4 cos^3 A - 3 cos A) = (3 tan A - tan^3 A)/(1- 3 tan^2 A)`

Prove the following trigonometric identities.

`cot theta - tan theta = (2 cos^2 theta - 1)/(sin theta cos theta)`

Prove the following trigonometric identities.

`tan theta - cot theta = (2 sin^2 theta - 1)/(sin theta cos theta)`

Prove the following trigonometric identities.

`(cos^2 theta)/sin theta - cosec theta +  sin theta  = 0`

if `sin theta = 3/4`  prove that `sqrt(cosec^2 theta - cot)/(sec^2 theta - 1) = sqrt7/3`

From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower

if `sec A = 17/8` verify that `(3 - 4sin^2A)/(4 cos^2 A - 3) = (3 - tan^2 A)/(1 - 3 tan^2 A)`

Prove the following trigonometric identities.

`1/(1 + sin A) + 1/(1 - sin A) =  2sec^2 A`

The angles of depression of the top and bottom of 8 m tall building from the top of a multistoried building are 30° and 45° respectively. Find the height of the multistoried building and the distance between the two buildings.

Prove the following trigonometric identities.

`(1 + sin theta)/cos theta + cos theta/(1 + sin theta) = 2 sec theta`

if `cot theta = 3/4` prove that `sqrt((sec theta - cosec theta)/(sec theta +cosec theta)) = 1/sqrt7`

A statue, 1.6 m tall, stands on a top of pedestal, from a point on the ground, the angle of elevation of the top of statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45 °. Find the height of the pedestal.

A T.V. Tower stands vertically on a bank of a river. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is 60°. From a point 20 m away this point on the same bank, the angle of elevation of the top of the tower is 30°. Find the height of the tower and the width of the river.

Prove the following trigonometric identities

`((1 + sin theta)^2 + (1 + sin theta)^2)/(2cos^2 theta) =  (1 + sin^2 theta)/(1 - sin^2 theta)`

If `tan theta = 24/7`, find that sin 𝜃 + cos 𝜃

From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower.

If `sin theta = a/b` find sec θ + tan θ in terms of a and b.

Prove the following trigonometric identities

`(1 + tan^2 theta)/(1 + cot^2 theta) = ((1 - tan theta)/(1 - cot theta))^2 = tan^2 theta`

If 8 tan A = 15, find sin A – cos A.

Prove the following trigonometric identities.

`(1 + sec theta)/sec theta = (sin^2 theta)/(1 - cos theta)`

As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

Prove the following trigonometric identities.

`tan theta/(1 - cot theta) + cot theta/(1 - tan theta) = 1 + tan theta + cot theta`

If 3cos θ – 4sin  = 2cos θ + sin θ Find tan θ

The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building

If `tan θ = 20/21` show that `(1 - sin theta + cos theta)/(1 + sin theta + cos theta) = 3/7`

From a point on a bridge across a river, the angles of depression of the banks on opposite side of the river are 30° and 45° respectively. If the bridge is at the height of 30 m from the banks, find the width of the river.

Prove the following trigonometric identities.

sec6θ = tan6θ + 3 tan2θ sec2θ + 1

Prove the following trigonometric identities

cosec6θ = cot6θ + 3 cot2θ cosec2θ + 1

If Cosec A = 2 find `1/(tan A) + (sin A)/(1 + cos A)`

Two poles of equal heights are standing opposite each other an either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30º, respectively. Find the height of poles and the distance of the point from the poles.

If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.

A man sitting at a height of 20 m on a tall tree on a small island in the middle of a river observes two poles directly opposite to each other on the two banks of the river and in line with the foot of the tree. If the angles of depression of the feet of the poles from a point at which the man is sitting on the tree on either side of the river are 60° and 30°respectively. Find the width of the river.

Prove the following trigonometric identities.

`((1 + tan^2 theta)cot theta)/(cosec^2 theta)   = tan theta`

Prove the following trigonometric identities.

`(1 + cos A)/sin^2 A = 1/(1 - cos A)`

If ∠A and ∠P are acute angles such that tan A = tan P, then show that ∠A = ∠P.

A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height 7m. At a point on the plane, the angle of elevation of the bottom of the flagstaff is 30º and that of the top of the flagstaff is 45º. Find the height of the tower.

Prove the following trigonometric identities.

`(sec A - tan A)/(sec A + tan A) = (cos^2 A)/(1 + sin A)^2`

In a ΔABC, right angled at A, if tan C = `sqrt3` , find the value of sin B cos C + cos B sin C.

The length of the shadow of a tower standing on the level plane is found to 2x meter longer when the sun's altitude is 30° than when it was 45°. Prove that the height of the tower is `x(sqrt3 + 1)` meters.

Prove the following trigonometric identities.

`(1 + cos A)/sin A = sin A/(1 - cos A)`

A tree breaks due to the storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with the ground. The distance from the foot of the tree to the point where the top touches the ground is 10 metres. Find the height of the tree.

State whether the following are true or false. Justify your answer.

The value of tan A is always less than 1.

State whether the following are true or false. Justify your answer.

sec A = 12/5 for some value of angle A.

State whether the following are true or false. Justify your answer.

cos A is the abbreviation used for the cosecant of angle A.

State whether the following are true or false. Justify your answer.

sin θ =4/3, for some angle θ

Prove the following trigonometric identities.

`sqrt((1 + sin A)/(1 - sin A)) = sec A + tan A`

A balloon is connected to a meteorological ground station by a cable of length 215 m inclined at 600 to the horizontal. Determine the height of the balloon from the ground. Assume that there is no slack in the cable.

Two men on either side of the cliff 80 m high observe the angles of an elevation of the top of the cliff to be 30° and 60° respectively. Find the distance between the two men.

Prove the following trigonometric identities.

`sqrt((1 - cos A)/(1 + cos A)) = cosec A - cot A`

`Prove the following trigonometric identities.

`(sec A - tan A)^2 = (1 - sin A)/(1 +  sin A)`

Find the angle of elevation of the sum (sun's altitude) when the length of the shadow of a vertical pole is equal to its height.

A fire in a building B is reported on the telephone to two fire stations P and Q, 20 km apart from each other on a straight road. P observes that the fire is at an angle of 60° to the road and Q observes that it is at an angle of 45° to the road. Which station should send its team and how much will this team have to travel?

Prove the following trigonometric identities.

`(1 - cos A)/(1 + cos A) = (cot A - cosec A)^2`

A man on the deck of a ship is 10 m above the water level. He observes that the angle of elevation of the top of a cliff is 45° and the angle of depression of the base is 300. Calculate the distance of the cliff from the ship and the height of the cliff.

Prove the following trigonometric identities.

`1/(sec A - 1) + 1/(sec A + 1) = 2 cosec A cot A`

A man standing on the deck of a ship, which is 8 m above water level. He observes the angle of elevation of the top of a hill as 60° and the angle of depression of the base of the hill as 30°. Calculate the distance of the hill from the ship and the height of the hill.

Prove the following trigonometric identities.

`cos A/(1 - tan A) + sin A/(1 - cot A)  = sin A + cos A`

There are two temples, one on each bank of a river, just opposite to each other. One temple is 50 m high. From the top of this temple, the angles of depression of the top and the foot of the other temple are 30° and 60° respectively. Find the width of the river and the height of the other temple.

Prove the following trigonometric identities.

`(cosec A)/(cosec A  - 1) + (cosec A)/(cosec A = 1) = 2 sec^2 A`

The angle of elevation of an aeroplane from a point on the ground is 45°. After a flight of 15 seconds, the elevation changes to 30°. If the aeroplane is flying at a height of 3000 metres, find the speed of the aeroplane.

Prove the following trigonometric identities.

`(1 + tan^2 A) + (1 + 1/tan^2 A) = 1/(sin^2 A - sin^4 A)`

Prove the following trigonometric identities.

`(tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1`

An aeroplane flying horizontally 1 km above the ground is observed at an elevation of 60°. After 10 seconds, its elevation is observed to be 30°. Find the speed of the aeroplane in km/hr.

From the top of a 50 m high tower, the angles of depression of the top and bottom of a pole are observed to be 45° and 60° respectively. Find the height of the pole.

Prove the following trigonometric identities.

`(cot A - cos A)/(cot A + cos A) = (cosec A - 1)/(cosec A + 1)`

The horizontal distance between two trees of different heights is 60 m. The angle of depression of the top of the first tree, when seen from the top of the second tree, is 45°. If the height of the second tree is 80 m, find the height of the first tree.

Prove the following trigonometric identities.

`(1 + cos theta + sin theta)/(1 + cos theta - sin theta) = (1 + sin theta)/cos theta`

Prove the following trigonometric identities.

`(sin theta - cos theta + 1)/(sin theta + cos theta - 1) = 1/(sec theta - tan theta)`

Prove the following trigonometric identities.

`(cos theta - sin theta + 1)/(cos theta + sin theta - 1) = cosec theta  + cot theta`

Prove the following trigonometric identities.

`1/(sec A + tan A) - 1/cos A = 1/cos A - 1/(sec A - tan A)`

A tree standing on a horizontal plane is leaning towards the east. At two points situated at distances a and b exactly due west on it,  the angles of elevation of the top are respectively α and β. Prove that the height of the top from the ground is `((b - a)tan alpha tan beta)/(tan alpha - tan beta)`

The angle of elevation of the top of a vertical tower PQ from a point on the ground is 60°. At a point Y, 40 m vertically above X, the angle of elevation of the top is 45°. Calculate the height of the tower.

Prove the following trigonometric identities

tan2 A + cot2 A = sec2 A cosec2 A − 2

Prove the following trigonometric identities.

`(1 - tan^2 A)/(cot^2 A -1) = tan^2 A`

The angle of elevation of a stationary cloud from a point 2500 m above a lake is 15° and the angle of depression of its reflection in the lake is 45°. What is the height of the cloud above the lake level? (Use tan 15° = 0.268)

Prove the following trigonometric identities.

`1 + cot^2 theta/(1 + cosec theta) = cosec theta`

Prove the following trigonometric identities.

`1 + cot^2 theta/(1 + cosec theta) = cosec theta`

If the angle of elevation of a cloud from a point h meters above a lake is a and the angle of depression of its reflection in the lake be b, prove that the distance of the cloud from the point of observation is `(2h sec alpha)/(tan beta - tan alpha)`

From an aeroplane vertically above a straight horizontal road, the angles of depression of two consecutive milestones on opposite sides of the aeroplane are observed to be α and β. Show that the height in miles of the aeroplane above the road is given by `(tan alpha tan beta)/(tan alpha + tan beta)`

Prove the following trigonometric identities.

`(cos theta)/(cosec theta + 1) + (cos theta)/(cosec theta - 1) = 2 tan theta`

PQ is a post of given height a, and AB is a tower at some distance. If α and β are the angles of elevation of B, the top of the tower, at P and Q respectively. Find the height of the tower and its distance from the post.

Prove the following trigonometric identities.

`(1 + cos theta - sin^2 theta)/(sin theta (1 + cos theta)) = cot theta`

Prove the following trigonometric identities.

`(tan^3 theta)/(1 + tan^2 theta) + (cot^3 theta)/(1 + cot^2 theta) = sec theta cosec theta - 2 sin theta cos theta`

A ladder rests against a wall at an angle α to the horizontal. Its foot is pulled away from the wall through a distance a so that it slides a distance b down the wall making an angle β with the horizontal. Show that `a/b = (cos alpha - cos beta)/(sin beta - sin alpha)`

 

Prove the following trigonometric identities.

if `T_n = sin^n theta + cos^n theta`, prove that `(T_3 - T_5)/T_1 = (T_5 - T_7)/T_3`

A tower subtends an angle 𝛼 at a point A in the plane of its base and the angle if depression of the foot of the tower at a point b metres just above A is β. Prove that the height of the tower is b tan α cot β

An observer, 1.5 m tall, is 28.5 m away from a tower 30 m high. Determine the angle of elevation of the top of the tower from his eye.

Prove the following trigonometric identities.

`[tan theta + 1/cos theta]^2 + [tan theta - 1/cos theta]^2 = 2((1 + sin^2 theta)/(1 - sin^2 theta))`

Prove the following trigonometric identities.

`(1/(sec^2 theta - cos theta) + 1/(cosec^2 theta - sin^2 theta)) sin^2 theta cos^2 theta = (1 - sin^2 theta cos^2 theta)/(2 + sin^2 theta + cos^2 theta)`

A carpenter makes stools for electricians with a square top of side 0.5 m and at a height of 1.5 m above the ground. Also, each leg is inclined at an angle of 60° to the ground. Find the length of each leg and also the lengths of two steps to be put at equal distances.

A boy is standing on the ground and flying a kite with 100 m of string at an elevation of 30°. Another boy is sanding on the roof of a 10 m high building and is flying his kite at an elevation of 45°. Both the boys are on opposite sides of both the kites. Find the length of the string that the second boy must have so that the two kites meet.

Prove the following trigonometric identities.

`((1 + sin theta - cos theta)/(1 + sin theta + cos theta))^2 = (1 - cos theta)/(1 + cos theta)`

The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of elevation of the top of the tower from the foot of the hill is 30°. If height of the tower is 50 m, find the height of the hill.

Prove the following trigonometric identities.

(sec A + tan A − 1) (sec A − tan A + 1) = 2 tan A

Two boats approach a lighthouse in mid-sea from opposite directions. The angles of elevation of the top of the lighthouse from two boats are 30° and 45° respectively. If the distance between two boats is 100 m, find the height of the lighthouse.

Prove the following trigonometric identities.

(1 + cot A − cosec A) (1 + tan A + sec A) = 2

Prove the following trigonometric identities.

(cosec θ − sec θ) (cot θ − tan θ) = (cosec θ + sec θ) ( sec θ cosec θ − 2)

From the top of a building AB, 60 m high, the angles of depression of the top and bottom of a vertical lamp post CD are observed to be 30° and 60° respectively. Find

1) the horizontal distance between AB and CD

2) the height of the lamp post.

3) the difference between the heights of the building and the lamp post.

From the top of a lighthouse, the angles of depression of two ships on the opposite sides of it are observed to be a and 3. If the height of the lighthouse be h meters and the line joining the ships passes through the foot of the lighthouse, show that the distance

`(h(tan alpha + tan beta))/(tan alpha tan beta)` meters

Prove the following trigonometric identities.

(sec A − cosec A) (1 + tan A + cot A) = tan A sec A − cot A cosec A

A straight highway leads to the foot of a tower of height 50 m. From the top of the tower, the angles of depression of two cars standing on the highway are 30° and 60° respectively. What is the distance the two cars and how far is each car from the tower?

Prove the following trigonometric identities.

`(cos A cosec A - sin A sec A)/(cos A + sin A) = cosec A - sec A`

The angles of elevation of the top of a rock from the top and foot of a 100 m high tower are respectively 30° and 45°. Find the height of the rock.

Prove the following trigonometric identities.

`sin A/(sec A + tan A - 1) + cos A/(cosec A + cot A + 1) = 1`

An observed from the top of a 150 m tall lighthouse, the angles of depression of two ships approaching it are 30° and 45°. If one ship is directly behind the other, find the distance between the two ships.

Prove the following trigonometric identities.

`tan A/(1 + tan^2  A)^2 + cot A/((1 + cot^2 A)) = sin A  cos A`

Prove the following trigonometric identities

sec4 A(1 − sin4 A) − 2 tan2 A = 1

A flag-staff stands on the top of a 5 m high tower. From a point on the ground, the angle of elevation of the top of the flag-staff is 60° and from the same point, the angle of elevation of the top of the tower is 45°. Find the height of the flag-staff.

Prove the following trigonometric identities.

`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`

The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m. from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is 6 m.

Prove the following trigonometric identities.

`(1 + cot A + tan A)(sin A - cos A) = sec A/(cosec^2 A) - (cosec A)/sec^2 A = sin A tan A - cos A cot A`

The angles of depression of two ships from the top of a lighthouse and on the same side of it are found to be 45° and 30° respectively. If the ships are 200 m apart, find the height of the lighthouse.

Prove the following trigonometric identities.

sin2 A cos2 B − cos2 A sin2 B = sin2 A − sin2 B

Prove the following trigonometric identities.

`(cot A + tan B)/(cot B + tan A) = cot A tan B`

Prove the following trigonometric identities.

`(tan A + tan B)/(cot A + cot B) = tan A tan B`

Prove the following trigonometric identities.

`cot^2 A cosec^2B - cot^2 B cosec^2 A = cot^2 A - cot^2 B`

Prove the following trigonometric identities.

tan2 A sec2 B − sec2 A tan2 B = tan2 A − tan2 B

Prove the following trigonometric identities

If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that x2 − y2 = a2 − b2

if `x/a cos theta + y/b sin theta = 1` and `x/a sin theta - y/b cos theta = 1` prove that `x^2/a^2 + y^2/b^2  = 2`

if `cosec theta - sin theta = a^3`, `sec theta - cos theta = b^3` prove that `a^2 b^2 (a^2 + b^2) = 1`

if `a cos^3 theta + 3a cos theta sin^2 theta = m, a sin^3 theta + 3 a cos^2 theta sin theta = n`Prove that `(m + n)^(2/3) + (m - n)^(2/3)`

Prove the following trigonometric identities.

if x = a cos^3 theta, y = b sin^3 theta` " prove that " `(x/a)^(2/3) + (y/b)^(2/3) = 1`

If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.

If a cos θ + b sin θ = m and a sin θ – b cos θ = n, prove that a2 + b2 = m2 + n2

If cos θ + cot θ = m and cosec θ – cot θ = n, prove that mn = 1

Prove the following trigonometric identities.

if cos A + cos2 A = 1, prove that sin2 A + sin4 A = 1

Prove that: `sqrt((sec theta - 1)/(sec theta + 1)) + sqrt((sec theta + 1)/(sec theta - 1)) = 2 cosec theta`

Prove that `sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta)) = 2 sec theta`

Prove that `sqrt((1 + cos theta)/(1 - cos theta)) + sqrt((1 - cos theta)/(1 + cos theta)) = 2 cosec theta`

Prove that `(sec theta - 1)/(sec theta + 1) = ((sin theta)/(1 + cos theta))^2`

If cos θ + cos2 θ = 1, prove that sin12 θ + 3 sin10 θ + 3 sin8 θ + sin6 θ + 2 sin4 θ + 2 sin2 θ − 2 = 1

Given that:
(1 + cos α) (1 + cos β) (1 + cos γ) = (1 − cos α) (1 − cos α) (1 − cos β) (1 − cos γ)

Show that one of the values of each member of this equality is sin α sin β sin γ

If sin θ + cos θ = x, prove that  `sin^6 theta + cos^6 theta = (4- 3(x^2 - 1)^2)/4`

If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z c tan θ, show that `x^2/a^2 + y^2/b^2 - x^2/c^2 = 1`

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if `cos theta = 4/5` find all other trigonometric ratios of angles θ

Evaluate the following

sin 45° sin 30° + cos 45° cos 30°

if `sin theta = 1/sqrt2`  find all other trigonometric ratios of angle θ.

Evaluate the following

sin 60° cos 30° + cos 60° sin 30°

if `tan theta = 1/sqrt2` find the value of `(cosec^2 theta - sec^2 theta)/(cosec^2 theta + cot^2 theta)`

Evaluate the following

cos 60° cos 45° - sin 60° ∙ sin 45°

if `tan theta = 3/4`, find the value of `(1 - cos theta)/(1 +cos theta)`

Evaluate the following

sin2 30° + sin2 45° + sin2 60° + sin2 90°

if `tan theta = 12/5` find the value of `(1 + sin theta)/(1 -sin theta)` 

Evaluate the following

cos2 30° + cos2 45° + cos2 60° + cos2 90°

Evaluate the following

tan2 30° + tan2 60° + tan45°

Evaluate the following

`2 sin^2 30^2 - 3 cos^2 45^2 + tan^2 60^@`

if `cosec A = sqrt2` find the value of `(2 sin^2 A + 3 cot^2 A)/(4(tan^2 A - cos^2 A))`

Evaluate the following

`sin^2 30° cos^2 45 ° + 4 tan^2 30° + 1/2 sin^2 90° − 2 cos^2 90° + 1/24 cos^2 0°`

if `cot theta = sqrt3` find the value of `(cosec^2 theta + cot^2 theta)/(cosec^2 theta - sec^2 theta)`

Evaluate the Following

4(sin4 60° + cos4 30°) − 3(tan2 60° − tan2 45°) + 5 cos2 45°

if `3 cos theta = 1`, find the value of `(6 sin^2 theta + tan^2 theta)/(4 cos theta)`

if `sqrt3 tan theta = 3 sin theta` find the value of `sin^2 theta - cos^2 theta`

Evaluate the following

(cosec2 45° sec2 30°)(sin2 30° + 4 cot2 45° − sec2 60°)

Evaluate the Following

cosec3 30° cos 60° tan3 45° sin2 90° sec2 45° cot 30°

Evaluate the Following

`cot^2 30^@ - 2 cos^2 60^2 - 3/4 sec^2 45^@ - 4 sec^2 30^@`

Evaluate the Following

(cos 0° + sin 45° + sin 30°)(sin 90° − cos 45° + cos 60°)

Evaluate the Following

`(sin 30^@ - sin 90^2 + 2 cos 0^@)/(tan 30^@ tan 60^@)`

Evaluate the Following

`4/(cot^2 30^@) + 1/(sin^2 60^@) - cos^2 45^@`

Evaluate the Following

4(sin4 30° + cos2 60°) − 3(cos2 45° − sin2 90°) − sin2 60°

Evaluate the Following

`(tan^2 60^@ + 4 cos^2 45^@ + 3 sec^2 30^@ + 5 cos^2 90)/(cosec 30^@ + sec 60^@ - cot^2 30^@)`

Evaluate the Following

`sin 30^2/sin 45^@ + tan 45^@/sec 60^@ - sin 60^@/cot 45^@ - cos 30^@/sin 90^@`

Evaluate the Following

`tan 45^@/(cosec 30^@) + sec 60^@/cot 45^@  - (5 sin 90^@)/(2 cos theta)`

Find the value of x in the following :

`2sin 3x = sqrt3`

Find the value of x in the following :

`2 sin  x/2 = 1`

Find the value of x in the following :

`sqrt3 sin x = cos x`

Find the value of x in the following :

tan 3x = sin 45º cos 45º + sin 30º

Find the value of x in the following :

`sqrt3 tan 2x = cos 60^@ + sin45^@ cos 45^@`

Find the value of x in the following :

cos 2x = cos 60° cos 30° + sin 60° sin 30°

If θ = 30° verify `tan 2 theta = (2 tan theta)/(1 - tan^2 theta)`

If θ = 30° verify that  `sin 2theta = (2 tan theta)/(1 + tan^2 theta)`

If 𝜃 = 30° verify `cos 2 theta = (1 - tan^2 theta)/(1 + tan^2 theta)`

f θ = 30°, verify that cos 3θ = 4 cos3 θ − 3 cos θ

If A = B = 60°, verify that cos (A − B) = cos A cos B + sin A sin B

If A = B = 60°, verify that sin (A − B) = sin A cos B − cos A sin B

If A = B = 60°. Verify `tan (A - B) = (tan A - tan B)/(1 + tan   tan B)`

If A = 30° B = 60° verify Sin (A + B) = Sin A Cos B + cos A sin B

If A = 30° and B = 60°, verify that cos (A + B) = cos A cos B − sin A sin B

If sin (A − B) = sin A cos B − cos A sin B and cos (A − B) = cos A cos B + sin A sin B, find the values of sin 15° and cos 15°.

In right angled triangle ABC. ∠C = 90°, ∠B = 60°. AB = 15 units. Find remaining angles and sides.

In ΔABC is a right triangle such that ∠C = 90° ∠A = 45°, BC = 7 units find ∠B, AB and AC

In rectangle ABCD AB = 20cm ∠BAC = 60° BC, calculate side BC and diagonals AC and BD.

If Sin (A + B) = 1 and cos (A – B) = 1, 0° < A + B ≤ 90° A ≥ B. Find A & B

If tan (A + B) = `sqrt3` and tan (A – B) = `1/sqrt3` ; 0° < A + B ≤ 90° ; A > B, find A and B.

If `sin (A – B) = 1/2` and `cos (A + B) = 1/2`, `0^@` < A + `B <= 90^@`, A > B Find A and B

In right angled triangle ΔABC at B, ∠A = ∠C. Find the values of Sin A cos C + Cos A Sin C

In right angled triangle ΔABC at B, ∠A = ∠C. Find the values of sin A sin B + cos A cos B

Find acute angles A & B, if sin (A + 2B) = `sqrt3/2 cos(A + 4B) = 0, A > B`

If A and B are acute angles such that tan A = 1/2,  tan B = 1/3 and tan (A + B) = `(tan A + tan B)/(1- tan A tan B)` A + B = ?

In ∆PQR, right-angled at Q, PQ = 3 cm and PR = 6 cm. Determine ∠P and ∠R.

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Evaluate the following:

`(sin 20^@)/(cos 70^@)`

Evaluate the following :

`cos 19^@/sin 71^@`

Evaluate the following :

`(sin 21^@)/(cos 69^@)`

Evaluate the following :

`tan 10^@/cot 80^@`

Evaluate the following

`sec 11^@/(cosec 79^@)`

Evaluate the following :

`((sin 49^@)/(cos 41^@))^2 + (cos 41^@/(sin 49^@))^2`

Evaluate cos 48° − sin 42°

Evaluate the following :

`(cot 40^@)/cos 35^@ -  1/2 [(cos 35^@)/(sin 55^@)]`

Evaluate the following :

`((sin 27^@)/(cos 63^@))^2 - (cos 63^@/sin 27^@)^2`

Evaluate the following :

`tan 35^@/cot 55^@  + cot 78^@/tan 12^@  -1`

Evaluate the following :

`(sec 70^@)/(cosec 20^@) + (sin 59^@)/(cos 31^@)`

Evaluate the following :

cosec 31° − sec 59°

Evaluate the following :

(sin 72° + cos 18°) (sin 72° − cos 18°)

Evaluate the following :

sin 35° sin 55° − cos 35° cos 55°

Show that tan 48° tan 23° tan 42° tan 67° = 1

Evaluate the following 

sec 50º sin 40° + cos 40º cosec 50º 

Express each one of the following in terms of trigonometric ratios of angles lying between
0° and 45°

Sin 59° + cos 56°

Express each one of the following in terms of trigonometric ratios of angles lying between 0° and 45°

tan 65° + cot 49°

Express each one of the following in terms of trigonometric ratios of angles lying between 0° and 45°

sec 76° + cosec 52°

Express each one of the following in terms of trigonometric ratios of angles lying between 0° and 45°

cos 78° + sec 78°

Express each one of the following in terms of trigonometric ratios of angles lying between 0° and 45°

cosec 54° + sin 72°

Express each one of the following in terms of trigonometric ratios of angles lying between 0° and 45°

cot 85° + cos 75°

Express each one of the following in terms of trigonometric ratios of angles lying between 0° and 45°

 sin 67° + cos 75°

Express cos 75° + cot 75° in terms of angles between 0° and 30°.

If Sin 3A = cos (A – 26°), where 3A is an acute angle, find the value of A =?

If A, B, C are the interior angles of a triangle ABC, prove that

`tan ((C+A)/2) = cot  B/2`

If A, B and C are interior angles of a triangle ABC, then show that `\sin( \frac{B+C}{2} )=\cos \frac{A}{2}`

Prove that  tan 20° tan 35° tan 45° tan 55° tan 70° = 1

Prove that sin 48° sec 42° + cos 48° cosec 42° = 2

Prove that `sin 70^@/cos 20^@  + (cosec 20^@)/sec 70^@  -  2 cos 20^@ cosec 20^@ = 0`

Prove that `cos 80^@/sin 10^@  + cos 59^@ cosec 31^@ = 2`

Prove the following

sin θ sin (90° − θ) − cos θ cos (90° − θ) = 0

Prove the following :

`(cos(90^@ - theta) sec(90^@ - theta)tan theta)/(cosec(90^@ - theta) sin(90^@ - theta) cot (90^@ -  theta)) + tan (90^@ - theta)/cot theta = 2`

Prove the following

`(tan (90 - A) cot A)/(cosec^2 A)   - cos^2 A =0`

Prove the following :

`(cos(90°−A) sin(90°−A))/tan(90°−A) - sin^2 A = 0`

Prove the following

 sin (50° − θ) − cos (40° − θ) + tan 1° tan 10° tan 20° tan 70° tan 80° tan 89° = 1

Evaluate: `2/3 (cos^4 30^@ - sin^4 45^@) - 3(sin^2 60^@ - sec^2 45^@) + 1/4 cot^2 30^@`

Evaluate: `4(sin^2 30 + cos^4 60^@) - 2/3 3[(sqrt(3/2))^2 . [1/sqrt2]^2] + 1/4 (sqrt3)^2`

Evaluate: `sin 50^@/cos 40^@ + (cosec 40^@)/sec 50^@  - 4 cos 50^@ cosec 40^@`

Evaluate tan 35° tan 40° tan 50° tan 55°

Evaluate: Cosec (65 + θ) – sec (25 – θ) – tan (55 – θ) + cot (35 + θ)

Evaluate: tan 7° tan 23° tan 60° tan 67° tan 83°

Evaluate: `(2sin 68)/cos 22 - (2 cot 15^@)/(5 tan 75^@) - (8 tan 45^@ tan 20^@ tan 40^@ tan 50^@ tan 70^@)/5`

Evaluate: `(3 cos 55^@)/(7 sin 35^@) -  (4(cos 70 cosec 20^@))/(7(tan 5^@ tan 25^@ tan 45^@ tan 65^@ tan  85^@))`

Evaluate: `sin 18^@/cos 72^@  + sqrt3 [tan 10° tan 30° tan 40° tan 50° tan 80°]`

Evaluate: `cos 58^@/sin 32^@  + sin 22^@/cos 68^@  - (cos 38^@ cosec 52^@)/(tan 18^@ tan 35^@ tan 60^@ tan 72^@ tan 65^@)`

If sin θ = cos (θ – 45°), where θ – 45° are acute angles, find the degree measure of θ

If A, B, C are the interior angles of a ΔABC, show that `cos[(B+C)/2] = sin A/2`

If 2θ + 45° and 30° − θ are acute angles, find the degree measure of θ satisfying Sin (20 + 45°) = cos (30 - θ°)

If θ is a positive acute angle such that sec θ = cosec 60°, find 2 cos2 θ – 1

If cos 2θ = sin 4θ where 2θ, 4θ are acute angles, find the value of θ.

If sin 3θ = cos (θ – 6°) where 3θ and θ − 6° are acute angles, find the value of θ.

If Sec 4A = cosec (A – 20°) where 4A is an acute angle, find the value of A.

If sec 2A = cosec (A – 42°) where 2A is an acute angle. Find the value of A.

R.D. Sharma 10 Mathematics

10 Mathematics

R.D. Sharma solutions for Class 10


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