Solution of Chapter 8. Introduction to Trigonometry and Its Applications (NCERT - Exemplar Mathematics Book)

If ., then the value of tan A is

If ., then the value of tan A is

If sin A=, then the value of cot A is

If sin A=, then the value of cot A is

The value of the expression cosec (75⁰+ θ)-sec (15⁰- θ)-tan (55⁰+ θ) + cot (35⁰- θ) is

The value of the expression cosec (75⁰+ θ)-sec (15⁰- θ)-tan (55⁰+ θ) + cot (35⁰- θ) is

If sin θ =, then cos θ is equal to

If sin θ =, then cos θ is equal to

If cos(α+β) = 0, then sin(α-β) can be reduced to

If cos(α+β) = 0, then sin(α-β) can be reduced to

The value of (tan 1°. tan 2°.tan 3° …… tan 89° ) is

The value of (tan 1°. tan 2°.tan 3° …… tan 89° ) is

If cos 9∝ =sin∝ and 9∝ <90⁰, then the value of tan 5∝ is

If cos 9∝ =sin∝ and 9∝ <90⁰, then the value of tan 5∝ is

If ABC is right angled at C, then the value of cos (A+B) is

If ABC is right angled at C, then the value of cos (A+B) is

If sin A+sin² A=1, then the value of (cos² A +cos⁴ A) is

If sin A+sin² A=1, then the value of (cos² A +cos⁴ A) is

If sin α = and cos β =, then the value of (∝ + β) is

If sin α = and cos β =, then the value of (∝ + β) is

The value of the expression

is

The value of the expression

is

If 4 tan θ =3, then is equal to

If 4 tan θ =3, then is equal to

If sin θ - cos θ = 0, then the value of (sin⁴ θ +cos⁴θ) is

If sin θ - cos θ = 0, then the value of (sin⁴ θ +cos⁴θ) is

sin (45⁰+θ) - cos (45⁰-θ) is equal to

sin (45⁰+θ) - cos (45⁰-θ) is equal to

If a pole 6 m high casts a shadow 2 m long on the ground then the sun’s elevation is

If a pole 6 m high casts a shadow 2 m long on the ground then the sun’s elevation is

=1

=1

The value of the expression (cos² 23⁰ - sin² 67⁰) is positive.

The value of the expression (cos² 23⁰ - sin² 67⁰) is positive.

The value of the expression (sin 80⁰-cos 80⁰) is negative.

The value of the expression (sin 80⁰-cos 80⁰) is negative.

= tan θ

= tan θ

If cos A+cos²A=1, then sin²A + sin⁴A=1

If cos A+cos²A=1, then sin²A + sin⁴A=1

(tan θ +2) (2 tan θ + 1) = 5 tan θ +sec² θ

(tan θ +2) (2 tan θ + 1) = 5 tan θ +sec² θ

If the length of the shadow of a tower is increasing then the angle of elevation of the sun is also increasing.

If the length of the shadow of a tower is increasing then the angle of elevation of the sun is also increasing.

If a man standing on a platform 3 m above the surface of a lake observes a cloud and its reflection in the lake, then the angle of elevation of the cloud is equal to the angle of depression of its reflection.

If a man standing on a platform 3 m above the surface of a lake observes a cloud and its reflection in the lake, then the angle of elevation of the cloud is equal to the angle of depression of its reflection.

The value of 2 sin can be a +, where a is a positive number and a1.

The value of 2 sin can be a +, where a is a positive number and a1.

, where a and b are two distinct numbers such that ab >0.

, where a and b are two distinct numbers such that ab >0.

The angle of elevation of the top of a tower is 30⁰. If the height of the tower is doubled, then the angle of elevation of its top will also be doubled.

The angle of elevation of the top of a tower is 30⁰. If the height of the tower is doubled, then the angle of elevation of its top will also be doubled.

If the height of a tower and the distance of the point of observation from its foot, both are increased by 10%, then the angle of elevation of its top remains unchanged.

If the height of a tower and the distance of the point of observation from its foot, both are increased by 10%, then the angle of elevation of its top remains unchanged.

+=2cosec θ

+=2cosec θ

= 2 cosec A

= 2 cosec A

If tan A=, then sin A. cos A=.

If tan A=, then sin A. cos A=.

(sin α + cos α) (tan α + cot α) = sec α + cosec α

(sin α + cos α) (tan α + cot α) = sec α + cosec α

(3-cot30⁰)=tan³60⁰-2sin60⁰

(3-cot30⁰)=tan³60⁰-2sin60⁰

1+

1+

tan θ +tan (90⁰- θ) = sec θ sec (90⁰- θ)

tan θ +tan (90⁰- θ) = sec θ sec (90⁰- θ)

Find the angle of elevation of the sun when the shadow of a pole h m high is h m long.

Find the angle of elevation of the sun when the shadow of a pole h m high is h m long.

If tan θ =1, then find the value of sin² θ -cos² θ.

If tan θ =1, then find the value of sin² θ -cos² θ.

A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60⁰ with the wall, then find the height of the wall.

A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60⁰ with the wall, then find the height of the wall.

Simplify (1+tan² θ)(1-sin θ)(1+sin θ)

Simplify (1+tan² θ)(1-sin θ)(1+sin θ)

If 2 sin² θ -cos² θ =2, then find the value of θ.

If 2 sin² θ -cos² θ =2, then find the value of θ.

Show that =1

Show that =1

An observer 1.5 m tall is 20.5 m away from a tower 22 m high. Determine the angle of elevation of the top of the tower from the eye of the observer.

An observer 1.5 m tall is 20.5 m away from a tower 22 m high. Determine the angle of elevation of the top of the tower from the eye of the observer.

Show that tan⁴ θ +tan² θ =sec⁴ θ -sec² θ.

Show that tan⁴ θ +tan² θ =sec⁴ θ -sec² θ.

If cosec θ +cot θ =p, then prove that cos θ =.

If cosec θ +cot θ =p, then prove that cos θ =.

Prove that =tan θ +cot θ.

Prove that =tan θ +cot θ.

The angle of elevation of the tower from certain point is 30⁰. If the observer moves 20 m towards the tower, the angle of elevation of the top increase by 15⁰.Find the height of the tower.

The angle of elevation of the tower from certain point is 30⁰. If the observer moves 20 m towards the tower, the angle of elevation of the top increase by 15⁰.Find the height of the tower.

If 1+sin²θ =3sinθ cos θ, then prove that tanθ =1 or .

If 1+sin²θ =3sinθ cos θ, then prove that tanθ =1 or .

If sin θ +2cos θ =1, then prove that 2 sin θ -cos θ =2.

If sin θ +2cos θ =1, then prove that 2 sin θ -cos θ =2.

The angle of elevation of the top of a tower from two points distant s and t from its foot are complementary. Prove that the height of the tower is .

The angle of elevation of the top of a tower from two points distant s and t from its foot are complementary. Prove that the height of the tower is .

The shadow of a tower standing on a level plane is found to be 50m longer when sun’s elevation is 30⁰ than when it is 60⁰. Find the height the tower.

The shadow of a tower standing on a level plane is found to be 50m longer when sun’s elevation is 30⁰ than when it is 60⁰. Find the height the tower.

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are α and β respectively. Prove that the height of the tower is .

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are α and β respectively. Prove that the height of the tower is .

If tan θ +sec θ =l, then prove that sec θ =

If tan θ +sec θ =l, then prove that sec θ =

If sin θ +cos θ =p and sec θ +cosec θ =q, then prove that q(p² - 1)=2p.

If sin θ +cos θ =p and sec θ +cosec θ =q, then prove that q(p² - 1)=2p.

If a sin θ +b cos θ =c, then prove that, a cos θ -b sin θ =

If a sin θ +b cos θ =c, then prove that, a cos θ -b sin θ =

Prove that =

Prove that =

The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60⁰ and the angle of elevation of the top of the second from the foot of the first tower is 30⁰. Find the top of distance between the two towers and also the height of the tower.

The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60⁰ and the angle of elevation of the top of the second from the foot of the first tower is 30⁰. Find the top of distance between the two towers and also the height of the tower.

From the top of a tower h m high, angle of depression of two objects, which are in line with the foot of the tower are α and β (β >α). Find the distance between the two objects.

From the top of a tower h m high, angle of depression of two objects, which are in line with the foot of the tower are α and β (β >α). Find the distance between the two objects.

A ladder rests against a vertical wall at an inclination α to the horizontal. Its foot is pulled away from the wall through a distance p, so that its upper end slides a distance q down the wall and then the ladder makes an angle to the horizontal, show that =.

A ladder rests against a vertical wall at an inclination α to the horizontal. Its foot is pulled away from the wall through a distance p, so that its upper end slides a distance q down the wall and then the ladder makes an angle to the horizontal, show that =.

The angle of elevation of the top of a vertical tower from a point on the ground is 60⁰.From another point 10 m vertically above the first its angle of elevation is 45⁰. Find the height of the tower.

The angle of elevation of the top of a vertical tower from a point on the ground is 60⁰.From another point 10 m vertically above the first its angle of elevation is 45⁰. Find the height of the tower.

A window of a house is h metres above the ground. From the window, the angles of elevation and depression of the top and the bottom of another house situated on the opposite side of the lane are found to be α and respectively. Prove that the height of the other house is h(1+tan α cot) metres.

A window of a house is h metres above the ground. From the window, the angles of elevation and depression of the top and the bottom of another house situated on the opposite side of the lane are found to be α and respectively. Prove that the height of the other house is h(1+tan α cot) metres.

The lower window of a house is at a height of 2 m above the ground and its upper window is 4 m vertically above the lower window. At certain instant the angles of elevation of a balloon from these windows are observed to be 60⁰ and 30⁰ respectively. Find the height of the balloon above the ground.

The lower window of a house is at a height of 2 m above the ground and its upper window is 4 m vertically above the lower window. At certain instant the angles of elevation of a balloon from these windows are observed to be 60⁰ and 30⁰ respectively. Find the height of the balloon above the ground.