# Ex. No. 4.1

1. Draw the figure and write the answers :

(i) For the angle in standard position if the initial arm rotates 220º in clockwise direction then terminal arm is in which quadrant ? [Ans.] [VIDEO]

(ii) For the angle in standard position if the initial arm rotates 25º in anticlockwise direction then terminal arm is in which quadrant? [Ans.] [VIDEO]

(iii) For the angle in standard position if the initial arm rotates 305º in anticlockwise direction then terminal arm is in which quadrant ? [Ans.] [VIDEO]

3. The terminal arm is on negative Y-axis, what are the possible angles ? What can you say about this angle ? [Ans.] [VIDEO]

1. Find the trigonometric ratios in standard position whose terminal arm passes through the points :

(ii) (5, – 12) [Ans.]

(iii) (– 24, – 7) [Ans.]

(v) (1, – 1) [Ans.]

(vi) (– 2, – 3) [Ans.]

3. Find where the angle lies if the terminal arm passes through :

(iv) (0, 2) [Ans.] [VIDEO]

4. If cos θ = 7/25 and θ is in the fourth quadrant, find the other five trigonometric ratios. [VIDEO]

4. If cos θ = 7/25 and θ is in the fourth quadrant, find the other five trigonometric ratios. [VIDEO]

Ex. No. 4.3

1. If sin θ = 5/13 . where θ is an acute angle, find the value of other trigonometric ratios using identities. [Ans.] [VIDEO]

2. If cot θ = - 7/24, then find the values of sin θ and sec θ, If θ is in IV quadrant. [Ans.] [VIDEO]

3. 3 sin α – 4 cos α = 0, then find the values of tan α , sec α and cosec α, where α is an acute angle. [Ans.] [VIDEO]

4. If tan θ = 1, then find the value of (sinθ + cos θ) ÷ (secθ + cosecθ ), where θ is an acute angle. [Ans.] [VIDEO]

5. If sec α = 2/√3, , then find the value of (1-cosec α)/(1+cosec α) , where α is in IV quadrant. [Ans.] [VIDEO]

6. Find the possible values of sin x if 8 sin x – cos x = 4. [Ans.] [VIDEO]

7. Prove the following :

7. Prove the following :

(ii) √((cosec x-1)/(cosec x+1))=1/(sec x+tan x) [Ans.]

(v) tanθ/(secθ+1)+(secθ+1)/tanθ = 2 cosecθ [Ans.]

(vi) (1+sinA)/cosA = (1+sinA+cosA)/(1+cos A-sinA ) [Ans.]

(vii) tanA/(secA-1) = (tanA+sec A+1)/(tan A+sec A-1) [Ans.]

(viii) √(sec

^{2}θ+cosec^{2}θ)= tanθ + cot θ [Ans.]
(ix) 1/(cosec A-cot A )-1/sinA=1/sinA-1/(cosec A+cot A) [Ans.]

(x) If tan A + 1/tan A = 2, Show that tan

^{2}A + 1/tan^{2}A = 2. [Ans.]
8. Eliminate θ, if

(i) x = a secθ, y = b tanθ [Ans.]

(ii) x = 2 cosθ – 3 sin θ , y = cos θ + 2 sin θ [Ans.]

(iii) x = 3 cosec θ + 4 cot θ, y = 4 cosec θ – 3 cot θ [Ans.]

Ex. No. 4.4

1. For a person standing at a distance of 80 m from a church, the angle of elevation of its top is of measure 45º. Find the height of the church. [Ans.] [VIDEO]

2. From the top of a lighthouse, an observer looks at a ship and find the angle of depression to be 60º. If the height of the lighthouse is 90 metres then find how far is that ship from the lighthouse ? ( √3 = 1.73). [Ans.] [VIDEO]

3. Two buildings are in front of each other on either side of a road of width 10 metres. From the top of the first building, which is 30 metres high, the angle of elevation of the top of the second is 45º. What is the height of the second building ? [Ans.] [VIDEO]

4. Two poles of height 18 metres and 7 metres are erected on the ground. A wire of length 22 metres tied to the top of the poles. Find the angle made by the wire with the horizontal. [Ans] [VIDEO]

5. A tree is broken by the wind. The top struck the ground at an angle of 30º and at a distance of 30 m from the root. Find the whole height of the tree. ( √3 = 1.73) [Ans.] [VIDEO]

6. A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to the ground. The inclination of the string with the ground is 60º. Find the length if the string, assuming that there is no slack in the string. ( √3 = 1.73) [Ans.] [VIDEO]

1. The angle of elevation of the top of a hill from the foot of a tower is 60 degree and the angle of elevation of the top of the tower from the foot of the hill is 30 degree. If the tower is 50 m high, then find the height of the hill.

2. A jet fighter at a eight of 3000 m from the ground, passes directly over another jet fighter at an instance when their angles of elevation from the same observation point are 60 degree and 45 degree respectively. Find the distance of the first jet fighter from the second jet at that instant.

3. The angle of elevation of an aeroplane from a point A on the ground is 60 degree. After a flight of 15 seconds horizontally, the angle of elevation changes to 30 degree. If the aeroplane is flying at a speed of 200 m/s, then find the constant height at which the aeroplane is flying.

4. A flag post stands on the top of a building. From a point on the ground, the angles of elevation of the top and bottom of the flag post are 60 degree and 45 degree respectively. If the height of the flag post is 10 m, find the height of the building.

5. A boy spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground level. The distance of his eye level from the ground is 1.2 m. The angle of elevation of the balloon from his eyes at an instant is 60 degree. After some time, from the same point of observation, the angle of elevation of the balloon reduces to 30 degree. Find the distance covered by the balloon during the interval.

6. A girl standing on a lighthouse build on a cliff near the seashore, observes two boats due East of the lighthouse. The angles of depression of the two boats are 30 degree and 60 degree. The distance between the boats is 300 m. Find the distance of the top of the lighthouse from the sea level.

7. A vertical wall and a tower are on the ground. As seen from the top of the tower, the angles of depression of the top and bottom of the wall are 40 degree and 60 degree respectively. Find the height of the wall if the height of the tower is 90 m .

8. A vertical tree is broken by the wind. The top of the tree touches the ground and makes an angle 30 degree with it. If the top of the tree touches the ground 30 m away from its foot, then find the actual height of the tree.

9. The angle of elevation of the top of a tower as seen by an observer is 30 degree. The observer is at a distance of 30√3m from the tower. if the eye level of the observer is 1.5m above the ground level, then find the height of the tower.

10. Find the angular elevation (angle of elevation from the ground level) of the sun when the length of the shadow of a 30m long pole is 10√3 m.

11. A ladder leaning against a vertical wall, makes an angle of 60 degree with the ground. The foot of the ladder is 3.5 m away from the wall. Find the length of the ladder.

12. A kite is flying with a string of length 200 m. If the thread makes an angle of 300 with the ground, find the distance of the kite from the ground level. (Here, assume the string is along a straight line)

13. If tan^2 α = cos^2 β – sin^2 β , then prove that cos^2α – sin^2 α = tan^2 β

14. tanθ + sin θ = m, tanθ – sinθ = n and m is not equal to n, then show that m^2-n^2=4√mn.

15. Prove the identity (cosecθ – sinθ )(secθ – cosθ ) = 1/(tanθ + cotθ)

16. Prove that (1+secθ)/secθ = sin^2θ/(1-cosθ)

17. Prove the identity [(secθ – tanθ )/(secθ +tanθ)]=1- 2secθ tanθ + 2tan^2 θ

18. Prove the identity [(sinθ – 2sin^3θ )/(2cos^3θ –cosθ )] = tanθ

19. Prove the identity (sin^6θ + cos^6θ ) = 1 – 3sin^2 θ cos^2 θ

20. Prove the identity (sinθ +cosecθ )^2 + (Cosecθ +secθ )^2 = 7+tan^2 θ + cot^2 θ

21. Prove the identity [tanθ /(1-cotθ)]+[cotθ /(1-tanθ )] = 1+tanθ +cotθ

22. Prove that [(tanθ + secθ – 1)/(tanθ – secθ +1)] = [(1+sinθ )/cosθ )]

23. Prove the identity [cosec (90 - θ ) – sin(90 - θ ) [cosec θ – sinθ ][tanθ + cotθ ] = 1

24. Prove the identity √(1-cosθ )/√(1+cosθ) = cosecθ – cotθ

25. Prove the identity (sinθ /cosecθ) + (cosθ /secθ ) = 1

**TRIGONOMETRY EXTRA HOTS FOR PRACTICE WITH SOLUTION**1. The angle of elevation of the top of a hill from the foot of a tower is 60 degree and the angle of elevation of the top of the tower from the foot of the hill is 30 degree. If the tower is 50 m high, then find the height of the hill.

2. A jet fighter at a eight of 3000 m from the ground, passes directly over another jet fighter at an instance when their angles of elevation from the same observation point are 60 degree and 45 degree respectively. Find the distance of the first jet fighter from the second jet at that instant.

3. The angle of elevation of an aeroplane from a point A on the ground is 60 degree. After a flight of 15 seconds horizontally, the angle of elevation changes to 30 degree. If the aeroplane is flying at a speed of 200 m/s, then find the constant height at which the aeroplane is flying.

4. A flag post stands on the top of a building. From a point on the ground, the angles of elevation of the top and bottom of the flag post are 60 degree and 45 degree respectively. If the height of the flag post is 10 m, find the height of the building.

5. A boy spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground level. The distance of his eye level from the ground is 1.2 m. The angle of elevation of the balloon from his eyes at an instant is 60 degree. After some time, from the same point of observation, the angle of elevation of the balloon reduces to 30 degree. Find the distance covered by the balloon during the interval.

6. A girl standing on a lighthouse build on a cliff near the seashore, observes two boats due East of the lighthouse. The angles of depression of the two boats are 30 degree and 60 degree. The distance between the boats is 300 m. Find the distance of the top of the lighthouse from the sea level.

7. A vertical wall and a tower are on the ground. As seen from the top of the tower, the angles of depression of the top and bottom of the wall are 40 degree and 60 degree respectively. Find the height of the wall if the height of the tower is 90 m .

8. A vertical tree is broken by the wind. The top of the tree touches the ground and makes an angle 30 degree with it. If the top of the tree touches the ground 30 m away from its foot, then find the actual height of the tree.

9. The angle of elevation of the top of a tower as seen by an observer is 30 degree. The observer is at a distance of 30√3m from the tower. if the eye level of the observer is 1.5m above the ground level, then find the height of the tower.

10. Find the angular elevation (angle of elevation from the ground level) of the sun when the length of the shadow of a 30m long pole is 10√3 m.

11. A ladder leaning against a vertical wall, makes an angle of 60 degree with the ground. The foot of the ladder is 3.5 m away from the wall. Find the length of the ladder.

12. A kite is flying with a string of length 200 m. If the thread makes an angle of 300 with the ground, find the distance of the kite from the ground level. (Here, assume the string is along a straight line)

13. If tan^2 α = cos^2 β – sin^2 β , then prove that cos^2α – sin^2 α = tan^2 β

14. tanθ + sin θ = m, tanθ – sinθ = n and m is not equal to n, then show that m^2-n^2=4√mn.

15. Prove the identity (cosecθ – sinθ )(secθ – cosθ ) = 1/(tanθ + cotθ)

16. Prove that (1+secθ)/secθ = sin^2θ/(1-cosθ)

17. Prove the identity [(secθ – tanθ )/(secθ +tanθ)]=1- 2secθ tanθ + 2tan^2 θ

18. Prove the identity [(sinθ – 2sin^3θ )/(2cos^3θ –cosθ )] = tanθ

19. Prove the identity (sin^6θ + cos^6θ ) = 1 – 3sin^2 θ cos^2 θ

20. Prove the identity (sinθ +cosecθ )^2 + (Cosecθ +secθ )^2 = 7+tan^2 θ + cot^2 θ

21. Prove the identity [tanθ /(1-cotθ)]+[cotθ /(1-tanθ )] = 1+tanθ +cotθ

22. Prove that [(tanθ + secθ – 1)/(tanθ – secθ +1)] = [(1+sinθ )/cosθ )]

23. Prove the identity [cosec (90 - θ ) – sin(90 - θ ) [cosec θ – sinθ ][tanθ + cotθ ] = 1

24. Prove the identity √(1-cosθ )/√(1+cosθ) = cosecθ – cotθ

25. Prove the identity (sinθ /cosecθ) + (cosθ /secθ ) = 1