CLASS X: CHAPTER - 8

INTRODUCTION TO TRIGONOMETRY

IMPORTANT FORMULAS & CONCEPTS

The word 'trigonometty' is derived from the Greek words 'tt·i' (meaning three), 'gon' (meaning sides) and
'metron' (meaning measure). In fact, higonometry is the study of relationships between the sides and angles
of a tt·iangle.

Trigonometric Ratios (T - Ratios) of an acute angle of a right triangle

In XOY-plane, let a revolving line OP staiting from OX, tt·ace out LXOP=0.
From P (x, y)draw PM 1- to OX.
In right angled triangle OMP. OM= x (Adjacent side); PM= y (opposite side); OP= r (hypotenuse).
y P(x, y)

X
X

y'
Opposite side y Hypotenuse r
sin 0= = cosec 0 =-------
Hypotenuse r Opposite side y
Adjacent Side x Hypotenuse !_
cosB= = secB= =
Hypotenuse r Adjacent Side x
Opposite side y Adjacent Side x
tan 0= = cot 0= =
Adjacent Side x Opposite side y

Reciprocal Relations
1 1
sin0=--- cosec0=-­
cosec0 sin0
1 1
cos0=-- sec0=--
sec0 cos0
1 1
tan0=-- cot0=--
cot0 tan0
Quotient Relations
sin0 cos0
tan 0 = -- and cot 0 =--
cos0 sin0
► Remark 1 : sin q is read as the "sine of angle q" and it should never be inte1preted as the product
of 'sin' and 'q'
► Remark 2 : Notation : (sin 0)2 is written as sin2 0 (read "sin square q") Similarly (sin et is
written as sinn e (read "sin nth po wer q" ), n being a positive integer.
► Note: (sin 0)2 should not be written as sin 02 or as sin2 02
► Remark 3 : T1igonometric ratios depend only on the value of 0 and are independent of the
lengths of the sides of the 1ight angled triangle.
 Trigonometric ratios of Complementary angles.
sin (90- 0) = cos 0 cos (90 - 0) = sin 0
tau (90 - 0) = cot 0 cot (90 - 0) = tau 0
sec (90 - 0) = cosec 0 cosec (90 - 0) = sec 0.

Trigonometric ratios for angle of measure.

o0, 30°,45°, 60° and 90° in tabular form.

LA Ou 30° 4 5° 60° 90°

sinA 0
1 1 ✓3 1
✓2
-
2 2
cosA 1 ✓3
-
1 1
- 0
2 ✓2 2
I
tanA 0
✓3
1 ✓3 Not defiued

cosecA Not defiued 2 ✓2 ✓3

1

secA 1
2 ✓2 2 Not defiued
✓3
I
cotA Not defiued ✓3 1
✓3
0

TRIGONOMETRIC IDENTITIES
Au equation involving trigonometric ratios of au angle is said to be a trigonometric identity if it is
satisfied for all values of 0 for which the given trigonometric ratios are defiued.
Identity (1): sin20 + cos20 = I
⇒ siti20 = 1- cos20 and cos20 = 1- sin20.
Identity (2) : sec2 0 = 1 + tau20
⇒ sec20- tau20 = I and tan20 = sec20- 1.
Identity (3) : cosec20 = 1 + cot20
⇒ cosec20 - cot20 = I and cot:20 = cosec20 - 1.
SOME TIPS
Coordinate System
Ri1ht Trianele SOH-CAH-TOA Method
Method
Opposite
SOH: sine(A)=sin(A)= sin(A)=t
Hypotenuse
Adjacent cos(A)=i
CAH: cos ine(A)= cos( A)
Hypotenuse
Hypotenuse /
Opposite Opposite tan( A)= 1'.'.
s;d•ly TOA: tangent(A)=tan(A)=
Side (y) Ad'Jacent

k� 90•1 cosecant (A) =csc(A) =-- =

1
sin(A)
Hypotenuse
OppoStte
1
csc(A)=--=-
h
sin(A) y
Adjacent Side (x) 1 h
1 Hypotenuse sec(A)=--=-
secant (A) =sec(A )=--= cos( A) x
cos(A) Adjacent
1
1 Adjacent cot(A)=- -=!!.
( ) =cot( A) =-- - =
cotangent A tan(A) y
tan A ( ) 0ppoStte
 Each trlgonomatrlc function In terms of the other five.
-
,-----f
in terms of sin(}
tan (J 1 Jsec'2 (J - 1 1
sin 8 = sin(} ±✓1 -cos20 ± Jl+tan 2 (J � ± sec() ±
Jl + cot20
cos0 = ± ✓i-sin2 0 cosO ±
1
Jl+tan 8 2 ±
Jes<? (J -1
csc 0
1
sec0
±
cotfJ
J1+cot2 0
sin0 J l _ cos20 1 ,----- t
tanlJ = ±
. ± ---- tan0 ±. l '2 ± J sec 2 0-1
v'1 -sin 0 2 cos 0 v ,csc 0-1 cot0
_1_ 1 Jl+tan (J 2 sec0 ,-----
cscO = ± ± cscO ± /1+ cot2 (J
sin 0 Jl -cos2 0 tan 0 Jsec 2 0- 1 ±
----
sec9= ±
1
. 2
v'l - SID 0
--
1
COS 0
± ✓
l+tan 2 ±
0 _,
�0
yCSC 28 - 1
s ec() ±
Jt+ cot20
cotO
J1 _ sin 0 2 cos 0 � 1 ,---- - 1
cotfJ=± ± 2 a: lJ ± Jcsc20-t± cotlJ
sinO Jt - cos 0 t n Jse c2 0 -1

Note: csc 0 is same as cos ec0.

 MCQ WORKSH££T= I
CLASSX:CHAPTER-8
INTRODUCTION TO TRIGONOMETRY

1. In !),.OPQ, right-angled atP, OP= 7 cm and O Q-PQ = 1 cm, then the values of sin Q.
7 24
(a) - (b) - (c) 1 (d) none of the these
25 25

24
2. If sin A= , then the value of cosA is
25
7 24
(a) - (b) - (c) 1 (d) none of the these
25 25

3. In !),.ABC, right-angled at B, AB= 5 cm and LACB = 30° then the length of the side BC is
(a) 5✓3 (b) 2✓3 (c) 10 cm (d) none of these

4. In !),. ABC, right-angled at B, AB= 5 cm and LACB = 30° then the length of the side AC is
(a) 5✓3 (b) 2✓3 (c) 10 cm (d) none of these

2 tan30°
5. The value of
1+tan 2300 IS
(a) sin 60° (b) cos 60° (c) tan 60° (d) sin 30°

1- tan 2 45 ° .
6. Tue value of
l+tan 2 450 IS
°
(a) tan 90 (b) 1 (c) sin 45 ° (d) 0

7. sin 2A= 2 sin A is trne when A=

(a) 0° (b) 30° (c) 45° (d) 60°

2 tan30°
8. Tue value of
1-tan 2 300 ts
(a) sin 60 °
(b) cos 60° (c) tan 60° (d) sin 30°

9. 9 sec2 A-9 tan2 A=

(a) 1 (b)9 (c) 8 (d) 0

10. (1 + tanA + secA ) (1 + cotA - cosecA ) =

(a) 0 (b) 1 (c) 2 (d)-1

11. (sec A+ tan A) (1 - sin A)=

(a) sec A (b) sin A (c) cosec A (d) cos A

1+ tru.i 2 A
12_ =
1+cot2 A
(a) sec2 A (b)-1 (c) cot2 A (d) tan2 A

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 MCQ WORKSH££T=U
CLASSX:CHAPTER-8
INTRODUCTION TO TRIGONOMETRY

1. If sin 3A= cos (A - 26 °), where 3A is an acute angle, find the value ofA
(a) 29° (b) 30° (c) 26 ° (d) 36 °

2. Iftan 2A= cot (A- 18°), where 2A is an acute angle, find the value ofA
(a) 29° (b) 30° (c) 26 ° (d) none ofthese

3. Ifsec 4A = cosec (A- 20°), where 4A is an acute angle, find the value ofA
(a) 22° (b) 25° (c) 26 ° (d) none ofthese

4. The value of tan 48° tan 23° tan 42° tan 67° is
(a) 1 (b) 9 (c) 8 (d) 0

5. If MBC is right angled at C, then the value ofcos(A + B) is

(a) 0 (b) 1 (c) ½ (d) n.d .

. sin 2 22 ° +sin 2 68° . 2 o o. o

6. The value ofthe expression [ 2 0 2 0
+ sm 63 + cos 63 sm 27 ]•is
cos 22 +cos 68
W3 MO �l �2

24
7. Ifcos A= , then the value ofsinA is
25
7 24
(a) - (b) (c) 1 (d) none ofthe these
25 25

8. If MBC is right angled at B, then the value ofcos(A + C) is

(a) 0 (b) 1 (c) ½ (d) n.d.

9. IftanA= ! , then the value ofcosA is

3
3
(a)- (c) 1 (d) none ofthe these
5

10. If MBC is right angled at C, in which AB= 29 units, BC= 21 units and LABC= a.
Detennine the values ofcos2a + sin2a is
(a) 0 (b) 1 (c) ½ (d) n.d.

11. In a right triangle ABC, right-angled at B, if tan A= 1, then the value of 2 sin A cos A=
(a) 0 (b) 1 (c) ½ (d) n.d.

12. Given 15 cot A= 8, then sin A=

3 4
(a)- (b) - (c) 1 (d) none ofthe these
5 3

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 MCQ WORKSH££T=IH
CLASSX:CHAPTER-8
INTRODUCTION TO TRIGONOMETRY

1. In a triangle PQR, right-angled at Q, PR+ QR= 25 cm and PQ= 5 cm, then the value of sin P is
7 24
(a) (b) (c) 1 (d) none of the these
25 25
2. In a triangle PQR, right-angled at Q, PQ= 3 cm and PR= 6 cm, then L'.QPR=
(a) 0° (b) 30° (c) 45° (d) 60°

3. If sin (A -B)= _!__ and cos(A+ B)= _!__, then the value of A and B, respectively ai-e
2 2
(a) 45° and 15 ° (b) 30 ° and 15° (c) 45° and 30° (d) none of these

4. If sin (A -B) = 1 and cos(A + B) = 1, then the value of A and B, respectively are
(a) 45° and 15 ° (b) 30 ° and 15° (c) 45° and 30° (d) none of these

5. If tan (A- B)= � and tan (A+ B)= ✓3 , then the value of A and B, respectively are
(a) 45° and 15 ° (b) 30° and 15° (c) 45° and 30° (d) none of these

6. If cos (A -B) = ✓3
and sin (A + B) = 1, then the value of A and B, respectively are
2
(a) 45° and 15 ° (b) 30 ° and 15° (c) 60° and 30° (d) none of these

7. The value of 2cos 2 60°+3sin 2 45° -3sin 2 30°+2cos 2 90° is

(a) 1 (b) 5 (c) 5/4 (d) none of these

8. sin 2A= 2 sin AcosA is true when A=

(a) 0° (b) 30° (c) 45° (d) any angle

9. sin A= cosA is true when A=

(a) 0° (b) 30° (c) 45° (d) any angle

10. If sinA = _!__ , then the value of 3cosA -4cos3A is

2
(a) 0 (b) 1 (c) ½ (d) n.d.

11. If 3cotA= 4, then the value of cos2A -sin2A is

3 7 24
(a) - (b) - (c) ½ (d)
4 25 25

3 sin A + 2 cos A
12. If 3tanA= 4, then the value of ----- is
3 sin A -2 cos A
24
(a) 1 (b) J_ (c) 3 (d)
25 25

.....................................................................................
 MCQ WORKSH££T=I\/
CLASSX:CHAPTER-8
INTRODUCTION TO TRIGONOMETRY

1. Value of 0, for sin 20 = 1, where o0 < 0 < 90° is:

(a) 30° (b) 60 ° (c) 45 ° (d) 135° .

2. Value of sec2 26° -cot2 64° is:

(a) 1 (b)-1 (c) 0 (d) 2

3. Product tanl 0.tan 2°.tan3°......tan89° is:

(a) 1 (b)-1 (c) 0 (d)90

4. ✓l+tan 0 is equal to:

2

(a) cot0 (b) cos0 (c) cosec0 (d) sec0

5. If A + B=90° , cot B= i then tanA is equal to;

4
(a) i4 (b) �
3
(c) _!_
4
(d) _!_
3

1
6. Maximmn value of , 0° < 0 < 90° is:
cosec0

(a) 1 (b)-1 (c) 2

7. If cos0 = _!_, sin¢= _!_ then value of 0 +¢ is

2 2
(a) 30° (b) 60 ° (c)90° (d) 120°.

8. If Sin (A+ B)= 1=cos(A- B)then

(a)A=B=90° (b)A=B=o0 (c)A=B=45° (d)A=2B

9. The value of sin60°cos30° -cos60°sin3o0 is

(a) 1 (b)-1 (c) 0 (d)none of these

10. The value of 2sin 2 30° -3cos 2 45°+tan2 60°+3sin 2 90° is

(a) 1 (b) 5 (c) 0 (d)none of these

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 PRACTICE QUESTIONS
CLASSX:CHAPTER-8
INTRODUCTION TO TRIGONOMETRY
TRIGONOMETRIC RATIOS

1 . cos ec2 0 - sec2 0

1. If tan0 = � ,what is the value of
-v 5 cos ec2 0 + sec2 0 ?

2. If sin 0 = i,
5
find the value of
sin 8 tan e- l
2
2tan 0
1 2secA
3. If cosA = - ,find the value of
2 l+tan 2 A

4. If sin 0 = .Jj ,find the value of all T- ratios of 0 .

2

5. If cos 0 = }_,find the value of all T- ratios of e .

25

6. If tane = � , find the value of all T- ratios of 0 .

8

7. If cot 0 = 2,find the value of all T- ratios of 0 .

8. If cosec 0 = .Jw ,find the value of all T- ratios of 0 .

9. If tan 0 = i,
3
show that (sin 0 + cos0) = 7_.
5
(sin0 - 2cos0) _ 12
5 ,sh ow that------
10 . If sec 0 =- -.
4 (tan0 - cot0) 7

1 (cosec2 0 - sec20) 3
11 . If tan =-,s
0 h ow that------=-
✓7 (cosec2 0 + sec2 0) 4

sine
12. If cos ec0 = 2, show that {cot0 +---} = 2 .
l+cos0

.
5 ,verify tan0 sin e
13. If sec 0 =- that 2
=
4 (1+tan 0) sec0

14. If cos 0 = 0.6, show that (5sin0 - 3tan0)=O.

15. In a triangle ACB, right-angled at C, in which AB = 29 units, BC = 21 units and L'.'ABC = 0.

Detennine the values of (i) cos2 0 + siii20 (ii) cos2 0 - sin2 0
16. In a triangle ABC, right-angled at B, in which AB = 12 cm and BC = 5cm Find the value of
cosA,cosecA, cosC and cosecC.
17. In a triangle ABC, L'.'B = 90°, AB = 24 cm and BC = 7 cm Find (i) SinA, CosA (ii) SinC, CosC.
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 PRACTICE QUESTIONS
CLASSX:CHAPTER-8
INTRODUCTION TO TRIGONOMETRY
I - RATIOS OF SOME PARTICULAR ANGLES
Evaluate each of the following:
1. sin 60° cos30° +cos 60° sin30°

2. cos 60° cos30° -sin 60° sin30 °

3. cos45° cos30° +sin45° sin30°

4. sin60° sin45° -cos60° cos45°

sin30° cot 45° sin60° cos30°

5• +
cos450 sec600 tan45° sin90°
tan2 60° +4cos2 45° +3cosec 2 60° + 2cos2 90°
6.
7
2cosec30° +3 sec60° --cot 2 30°
3
7. 4(sin4 30° +cos4 60° )-3(cos2 45° -sin2 90° ) + 5 cos2 90°

8. ; +
cot 300 sin 300
}
2 cos2 45 ° -sin2 o0

1 1
9• +
cos 30
2 °
sin 30°
3

3 1
10. cot 2 30° -2 cos 2 30° --sec2 45 ° +-cos ec2 30°
4 4

11. (sin2 30° +4cot2 45° -sec2 60° )(cosec 2 45° sec2 30° )

12. In right triangle ABC , LB= 90°, AB= 3cm and AC= 6cm Find LC and LA.

13. If A= 30°, verify that:

. . 2tanA c·· 1-tan2 A c··· 2 tanA
( 1 ) sm2A= 11 ) cos 2A= m) tan2A=
l+tan A
2
l+ tan A
2
1-tan2 A

14. If A= 45°, verify that

( i) sin2A= 2sinAcosA ( ii) cos2A = 2cos2A- I= I - 2sin2A

✓
15. Using the fo1mula, cos A = l
+cos 2A
, find the value ofcos30° , it be ing given that cos60° = .!._
2 2

=✓
l-cos 2A . 1
• t he fc01mula, smA
16. Usmg • , find the va 1ue o f sin30
• 0
• •
, 1t be mg given t hat cos60 = -
O
2 2
 2tan:
17. Using the fonnula, tan 2A = , find the value oftan60°, it being given that
I-tan A
1
tan300 = ✓3 .

18. Ifsin (A- B)= .!_ and cos(A+ B) = .!_, then find the value ofA and B.
2 2
19. Ifsin (A+ B)= 1 and cos(A- B)= 1, then find the value ofA and B.
20. Iftan ( A- B) = � and tan ( A+ B) = ✓3 , then find the value ofA and B.

21. Ifcos (A- B) = ✓3 and sin (A+ B) = 1, then find the value ofA and B.
2
1 1 tanA+tanB
22 . IfA andB are acute angles sueh that tanA = - , tanB = - and tan(A +B)= -----,
3 2 1-tanAtanB
show that A+ B = 45 °.

23. IfA= B= 45°, verify that:

a) sin(A+ B)= sinAcosB+ cosAsinB
b) sin(A- B) = sinAcosB- cosAsinB
c) cos(A + B)= cosAcosB - sinAsinB
d) cos(A- B) = cosAcosB+ sinAsinB
tanA+tanB
e) tan(A+B)=
1-tanAtanB
f) tan(A-B)= tanA-tanB
l+tanAtanB

24. IfA= 60° and B= 30°, verify that:

a) sin(A+ B) = sinAcosB+ cosAsinB
b) sin(A- B) = sinAcosB- cosAsinB
c) cos(A+ B)= cosAcosB- sinAsinB
d) cos(A- B)= cosAcosB+ sinAsinB
tanA+ tanB
e) tan(A+B)=
1-tanAtanB
tanA-tanB
f) tan ( A-B )=----­
l+tanAtanB
25. Evaluate:
sin2 45 °+ i cosec230° - cos 60° +tan2 60°
4

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 PRACTICE QUESTIONS
CLASSX:CHAPTER-8
INTRODUCTION TO TRIGONOMETRY
I - RATIOS OF COMPI,EMENTARY ANGLES

1. Evaluate: cot0 tan(90° - 0) - sec (90° - 0) cosec0+ (sii/25° + sin265°) + ✓3 (tan5°. tan15°.
tan30°. tan75°. tan85° ).

sec0cosec(90°-0)-tan 0cot(90°-0)+ (sin 2 35° +sin 2 55° )

2. Evaluate without using tables:
tan 10° tan 20° tan 45° tan 70° tan 80°
--------c----,---- ----,--- -,-------=-----

sec 2 54 ° -cot2 36° . 2 o

3. Evaluate: s
0 +2 rn 38 sec 52
2 o
-srn
. 2
45 .
o
cosec 57 -tan 33
2 0 2

4. Express sin67° + cos75° in terms oftrigonometric ratios ofangles between o0 and 45°.

5. Ifsin4A = cos(A-20° ), where A is an acute angle, find the value ofA

B C
6. IfA, B and C are the interior angles oftriangle ABC, prove that tan( � ) = cot �

7. IfA, B, C ai·e interior angles ofa MBC, then show that co{ B�C) =sin i .

8. IfA, B, C ai·e interior angles ofa MBC, then show that co s ec( A; C) =sec!.
9. IfA, B, C are interior angles ofa MBC, then show that co{ B;A)= tan�.

c s 70:
10. Without using trigonometric tables, find the value of � +cos57° cosec33° -2cos60° .
Slll 20

11. Ifsec4A = cosec(A-20° ), where 4A is an acute angle, find the value ofA

12. Ifta11 2A = cot (A- 40°), where 2A is an acute angle, find the value ofA

13. Evaluate tanl 0°tanl 5°tan75°tan80°

[ S111
• 2 • 2 6 80
220 +S111 . 2 o
14. Evaluate: 0 +sm 63 +cos 63 s11127
O . O

cos 22 +cos 6 8
2 0 2
]
15. Express tan60° + cos46° in tenns oftrigonometric ratio s ofangles between 0° and 45°.

16. Express sec51° + cosec25° in terms oftrigonometiic ratios ofangles between o0 and 45°.

17. Express cot77° + sin54° in terms oftiigonometi·ic ratios of angles between 0° and 45°.

18. Iftan 3A = cot (3A-60° ), where 3A is a11 acute angle, find the vah1e ofA

19. Ifsin2A = co s(A+ 36°), where 2A is an acute angle, find the value ofA

20. IfcosecA = sec(A-10° ), where A is an acute angle, find the value ofA
21. Ifsin50 = cos40, where 50 and 40 are acute angles, find the value of0.

22. Iftan 2A = cot (A- 18°), where 2A is an acute angle, find the value ofA

23. Iftan20 = cot(0 + 6°), where 20 and 0 + 6° are acute angles, find the value of0.

24. Evaluate:
2 sin 68° 2 cot 15° 3 tan 45° tan 20° tan 40° tan 50° tan 70°
cos 22° 5 tan75° 5
25. Evaluate:
cos(90° -0)sec(90° -0)tan0 tan(90° -0)
+----+2
cos ec(900 -0) sin(900 - 0) cot(900 - 0) cot 0

26. Evaluate:
sin 18: ../3
+ {tan 10° tan 30° tan 40° tan 50° tan 80° }
cos72
27. Evaluate:
3cos55° 4(cos70° cos ec20° )
7sin 35° ?(tan 5° tan 25° tan 45° tan 65° tan 85° )

28. Evaluate:
0 _ . 0 cos 2 40° +cos 2 50°
COS (40 0)-Sill(50 +0)+ 2 o
sin 40 +sin 2 50o

tan A tan B+tan A cot B _ sin 2 B

29. IfA + B = 90°, prove that = tan A
sinAsecB cos 2 A

30. Ifcos20 = sin40, where 20 and 40 are acute angles, find the value of0.
 PRACTICE QUESTIONS
CLASSX:CHAPTER-8
INTRODUCTION TO TRIGONOMETRY
TRIGONOMETRIC IDENTITIES
cose sine .
1. Provethat ---+---=sme+ cose.
1- tan e 1- cote

1 sin0 l+ cos0 1
2. Provethat - {---+--- } =--.
2 1+ cos0 sin 0 sin 0

tan3 a cot 3 a
3. Provethat: --- + =secacoseca-2smacosa
l+ tan2 -
a l+cot 2 a
tan A cot A
4. Provethat: ---+ --- = 1+ tan A+cot A = 1+ sec Acos ecA .
1-cot A 1- tan A

l+sinA-cosA 1-cosA
5. Provethat:.- - - - -- =
1+sin A+cos A 1+ cos A

6. Provethta (tan A+ cosecB)2-(cotB- secA)2= 2tanA cot B (cosecA+ secB).

cos A I+ sin A
7. Provethat: ---+ --- = 2 sec A
1+ sin A cos A

cos A - sin A+1

8. Provethta : = cos ecA+ cot A .
cos A+ sin A-1
sin A+ cos A sin A- cos A 2 2
= --2 -
9. Provethat:-- - - + = 2 = -
sinA- cosA sinA+cosA sin A-cos A 2sin A-1 l-2 cos 2 A·
2 2

sin A sin A
10. Provethat = 2+-----
cot A+ cosecA cotA-cosecA

secA-1 secA+l
11. Provethat ,---+ --- = 2 cosec:A .
sec A+ 1 secA-1

1 1 1 1
12.Provethat: ------ --=--------
cos ecA- cotA sin A sin A cos ecA+ cot A

tan e+sece-1 1+sine

13.Provethat: ----- = sece+tan e= --
tan e -sece+l cose

14.Ifx = asin0+bcos0 an dy= acos0+bsin 0,provethat x2 +y2 =a2 +b2 .

11l-1
15. If sec0+tan0 = m, show that (-2-· ) = sin0.
m +l
 l+cos0+sin0 _ l+sin0
n..
16. .nove that.. ----------.
1+cos 0-sin 0 cos 0
17. Prove that sec4 A(l-sin A)-2tan 2 A=1
4

18.Ifcosec0-sin0=m and sec0-cos0=n,prove that ( m 2n) / +(mn2) =1

23 2/3

19.Iftan0+sin0 = m and tan0-sin0 =n, show that m2-n 2 =�

20.Ifacos0-bsin0 =c, prove that (a sin0+bcos0) =± a 2+b2-c2 ✓

21.Ifcos0+ sin0 = ✓2 cos0, prove that cos0-sin0 = ✓2 sin0

22.If(�sine-; cos0J=l and (�cose+: sin0J=l,prove that (;: +;:)=2

23.If(tan0+sin0)=m and (tan0-sin0)=n prove that (m2 -n2) =16mn

2

24.Ifcosec0-sin0 =a3 and sec0-cos0 =b3, prove that a 2b2(a 2 +b2)=1

25.Ifacos 3 0+3a sin 2 0 cos0 =m and asin3 0+3asin0 cos 2 0 =n, prove that
2/3 2/3
( m+n ) +(m-n) =2a213

✓
26. Prove that sec 2 0+cosec 20 =tan0+cot0.

sin0-cos0+1 1
27. Prove the identity:
sin 0+ cos 0 - 1 sec 0 - tan 0

28. Prove the identity: sec6 0 = tan6 0 +3 tan2 0.sec2 0 +I.

29. Prove the identity: (sin A+ cosec A)2 + (cos A+ secA)2 =7 + tan2A+ cot 2A.

30.Ifxsin3 0+ycos3 0 =sin0cos0 and xsin0 =ycos0, prove that x2 +y 2=1.

1 1
31.Ifsec0 = x +-,Prove that sec0 +tan 0= 2x or -.
4x 2x

32. Prove that (1+ \ )(1+-1--) = . 1

. •
tan A cot A
2
sm A-sm 4 A
2

. cot0 + tan0 = x and sec0 - cos0 = y, prove that ( x 2 y) -( -'.91 2)

33If =1_
213 2/3

cos a cos a
34.If--=m and -.-=n ' show that (m 2 +n 2 ) cos2 /3=11 2
cos/3 sm/3

35.Ifcosec0-sin0 = a and sec0-cos0 = b, prove that a2b2(a2 + b2 +3)

 r
36. Ifx = rsinAcosC, y = rsinAsinC and z = rcosA, prove that r2 = +-/+ 'C.

.
. = msinB, ni2 -1
37. IftanA = n tanB and sinA prove that cos2 A= - -.
n2 -1

38. Ifsine+ sin2e = 1, find the value ofcos12e+ 3cos 10e+ 3cos8e+ cos6e+ 2cos4e+ 2cos2e- 2..

39. Prove that: (1-sine+ cose)2 = 2(1+ cose)(l -sine)

40. Ifsin+ sin2e = 1, prove that cos2e+ cos4e = 1.

41. Ifasece+ btan0+ c = 0 and psec0+ qtane+ r = 0, prove that (br-qc/-(pc-ar) 2 = (aq-bp)2 .

42. Ifsine+ sin20+ sin3 e = 1, then prove that cos60-4cos40+ 8cos2e = 4.

43. Iftan20 = 1 -a2, prove that sece+ tan30cosec0 = (2-a2)312 .

44. Ifx = asec0+ btan0 and y = atan0+ bsece, prove that '2---/ = a2 - b2 .

45. If3sin0+ 5cos0 = 5, prove that 5sin0- 3cose = ± 3.

.....................................................................................
 CLASSX:CHAPTER-9
SOME APPLICATIONS TO TRIGONOMETRY

IMPORTANT FORMULAS & CONCEPTS

ANGLE OF ELEVATION
In the below figure, the line AC drawn from the eye of the student to the top of the minar is called
the line of sight. The student is looking at the top of the minar. The angle BAC, so formed by the line
of sight with the horizontaL is called the angle of elevation of the top of the minar from the eye of
the student. Thus, the line of sight is the line drawn from the eye of an obse1ver to the point in the
object viewed by the obse1ver.
C
Object

·. 'i<..����� -�� �-1��:��i�-� ...........

A
E f ...................................
The angle of elevation of the point viewed is the angle fonned by the line of sight with the
horizontal when the point being viewed is above the horizontal level, i.e., the case when we raise our
head to look at the object

ANGLE OF DEPRESSION
In the below figure, the girl sitting on the balcony is looking down at a flower pot placed on a stair of
the temple. In this case, the line of sight is below the horizontal level. The angle so formed by the
line of sight with the horizontal is called the angle of depression. Thus, the angle of depression of a
point on the object being viewed is the angle fo1med by the line of sight with the horizontal when the
point is below the horizontal leveL i.e., the case when we lower our head to look at the point being
viewed

- - -Horizontal
- - - - - -----level
- - - - - - - - ---- - --- - ·
.AngJeofde
pressfon

".

Trigonometric Ratios (T - Ratios) of an acute angle of a right triangle

In XOY-plane, let a revolving line OP starting from OX, trace out LXOP=0. From P (x, y)draw PM
.L to OX.
In right angled triangle OMP. OM = x (Adjacent side); PM= y (opposite side); OP = r (hypotenuse).
 y P(x, y)

X
X

y'
pposite Side = y Adjacent Side � posite Side y
sin 0 = O cos0 = = tan0 = Op =
Hypotenuse r ' Hypotenuse r' Adjacent Side x
Hypotenuse !__ Hypotenuse !__ Adjacent Side �
cos ece = = sec0 = = cot0 = =
Opposite Side y ' Adjacent Side x ' Opposite Side y

Reciprocal Relations
1 1 1
cos ec0 = -- , sec0 = -- and cot 0 = --
sin 0 cos0 tan0

Quotient Relations
sin0 cos0
tan0 =-- and cot0=--
cos0 sin0

Trigonometric ratios of Complementary angles.

sin (90 - 0) = cos 0 cos (90 - 0) = sin 0
tan (90 - 0) = cot 0 cot (90 - 0) = tan 0
sec (90 - 0) = cosec 0 cosec (90 :-- 0) = sec 0.

Trigonometric ratios for angle of measure.

o0, 30°,45°, 60° and 90° in tabular form.

LA Ou 30° 45° 60° 90°

sinA 0
1 1 ✓3 1
✓2
-
2 2

cosA 1 ✓3
-
1 -
1
0
2 ✓2 2
1
tanA 0
✓3
1 ✓3 Not defined

cosecA Not defined 2 ✓2 ✓3

1
2
secA 1
✓3 ✓2 2 Not defined

1
cotA Not defined ✓3 1
✓3
0

•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
 MCQ WORKSH££T= I
CLASS X: CHAPTER - 9
SOME APPLICATIONS TO TRIGONOMETRY
1. The angle of elevation ofthe top ofa tower from a point on the grmmd, which is 20m away from
the foot 3ofthe tower is 60°3. Find the height ofthe tower.
3
(a) 10 ✓ m (b) 30 ✓ m (c) 20 ✓ m (d) none ofthese

2. The height ofa tower is 10m What is the length ofits shadow when Sun's altitude is 45°?
(a) 10 m (b) 30 m (c) 20 m (d) none ofthese

3. The angle ofelevation ofa ladder leaning against a wall is 60° and the foot ofthe ladder is 9.5 m
away from the wall. Find the length ofthe ladder.
(a) 10 m (b) 19 m (c) 20 m (d) none ofthese
3
4. Ifthe ratio ofthe height ofa tower and the length ofits shadow is ✓ : 1, what is the angle of
elevation ofthe Sun?
(a) 30° (b) 60° (c) 45° (d) none ofthese
5. What is the angle ofelevation ofthe Sm1 when the length ofthe shadow ofa ve1iical pole is
equal to its height?
(a) 30° (b) 60° (c) 45° (d) none ofthese

6. From a point on the ground, 20 m away from the foot ofa ve1iical tower, the angle ofelevation
°
ofthe top
3 ofthe tower is 60 , what is the height ofthe tower?
(a) 10 ✓ m (b) 30 ✓3 m (c) 20 ✓3 m (d) none ofthese
7. Ifthe angles ofelevation ofthe top of a tower from two points at a distance of4 m and 9 m from
the base ofthe tower and in the same straight line with it are complementaiy, find the height of
the tower.
(a) 10 m (b) 6 m (c) 8 m (d) none ofthese
8. In the below fig. what are the angles ofdepression from the observing positions D and E ofthe
object A?
(a) 30°, 45° (b) 60° ' 45° (c) 45°, 60° (d) none ofthese
D

C B
9. The ratio ofthe length ofa rod and its shadow is 1: ✓3 . The angle ofelevation ofthe Slm is
(a) 30° (b) 60° (c) 45° (d) none ofthese
10. Ifthe angle ofelevation ofa tower from a distance of 100m from its foot is 60°, then the height
ofthe tower is
200 r:; m (d) 100
(a) 100 ✓3 m (b) 3 m (c) 50-v-' 3 m
✓ ✓
 MCQ WORKSH££T=U
CLASS X: CHAPTER - 9
SOME APPLICATIONS TO TRIGONOMETRY

1. If the altitude of the sun is at 60°, then the height of the ve1tical tower that will cast a shadow of
length 30m is
} 30
(a) 30 ✓ m (b) 15 m (c) m (d) 15✓2 m
✓}
2. A tower subtends an angle of 30° at a point on the same level as its foot. At a second point 'h'
metres above the first, the depression of the foot of the tower is 60°. The height of the tower is
h h h m h
(a) - m (b) - m (c) ..;3h 3
(d) -m
2
3
3 ✓
3. A tower is 100 ✓ m high. Find the angle of elevation if its top from a point 100 m away from its
foot.
(a) 30° (b) 60° (c) 45° (d) none of these

4. The angle of elevation of the top of a tower from a point on the grotmd, which is 30m away from
the foot 3of the tower is 30°3. Find the height of the tower.
(a) 10 ✓ m (b) 30 ✓ m (c) 20 ✓3 m (d) none of these

5. The string of a kite is 100m long and it makes an angle of 60° with the horizontal. Find the
height of the kite, assuming that there is no slack in the string.
200
(a) 100 ✓3 m (b) m (c) 50 ✓3 m (d) lOO m
.Jj .Jj
6. A kite is flying at a height of 60m above the ground. The inclination of the string with the ground
is 60 °. Find the length of the string, assuming that there is no slack in the string.
3 3 3
(a) 40 ✓ m (b) 30 ✓ m (c) 20 ✓ m (d) none of these

7. A circus rutist is climbing a 20m long rope, which is tightly stretched and tied from the top of a
ve1tical pole to the ground. Find the height of the pole if the angle made by the rope with the
ground level is 30°.
(a) 10 m (b) 30 m (c) 20 m (d) none of these

8. A tower is 50m high, Its shadow ix 'x' metres shmter when the sun's altitude is 45° than when it
is 30°. Find the value of 'x'
200
(a) 100 ✓3 m (b) m (c) 50 ✓3 m (d) none of these
.Jj 3
9. Find the angular elevation of the sun when the shadow of a 10m long pole is 10 ✓ m
(a) 30° (b) 60° (c) 45° (d) none of these

10. A vettical pole stands on the level ground. From a point on the ground 25m away from the foot
of the pole, the ru1gle of elevation of its top is found to be 60°. Find the height of the pole.
(a) 25 ✓3 m (b) f3 m (c) 50 ✓3 m (d) none of these
 MCQ WORKSH££T=IH
CLASS X: CHAPTER - 9
SOME APPLICATIONS TO TRIGONOMETRY

1. A kite is flying at a height of75m above the grnund. The inclination of the string with the ground
is 60°. Find
3 the length of the 3 string, assuming that there is no slack in the string.
(a) 40 ✓ m (b) 30 ✓ m (c) 50 ✓3 m (d) none of these
2. The angle of elevation ofthe tope ofa tree from a point A on the ground is 60°. On wallcing 20m
away from its base, to a point B, the angle of elevation changes to 30°. Find the height of the
tree.
(a) 10 ✓3 m (b) 30 ✓3 m (c) 20 ✓3 m (d) none of these

3. A 1.5m tall boy stands at a distance of2m from lamp post and casts a shadow of4.5m on the
ground. Find the height of the lamp post.
(a) 3 m (b) 2.5 m (c) 5 m (d) none of these

4. The height of the tower is 100m. When the angle of elevation of the sun changes from 30° to 45°,
the shadow of the tower becomes 'x' meters less. The value of 'x' is
100
(a) 100 ✓3 m (b)100 m (c) 100( ✓3 - 1) m (d)
✓3
5. The tops of two poles of height 20m and 14m are collllected by a wire. If the wire makes an
angle of 30° with horizontal, then the length of the wire is
(a) 12 m (b)10 m (c) 8 m (d) 6 m

6. If the angles of elevation of a tower from two points distant a and b (a > b) from its foot and in
the same straight line from it are 30° and 60°, then the height of the tower is

(a) ../a+b m (b) ../a-b m (c) Mm (d) � m

7. The angles of elevation of the top of a tower from two points at a distance of 'a' m and 'b' m
from the base of the tower and in the same straight line with it are complementaiy, then the
heigl1t of the tower is

(a) ../a+b m (b) ../a-b m (c)M m

8. From the top of a cliff 25m high the angle of elevation ofa tower is found to be equal to the
angle of depression of the foot ofthe tower. The height of the tower is
(a) 25 m (b) 50 m (c) 75 m (d) 100 m

9. If the angle of elevation of a cloud from a point 200m above a lake is 30° and the angle of
depression of its reflection in the lake is 60°, then the height of the cloud above the lake is
(a) 200 m (b) 500 m (c) 30 m (d) 400 m

10. The angle of elevation of a cloud from a point 'h' meter above a lake is 'a'. The angle of
depression ofits reflection in the lake is 45°. The height of the cloud is
(a) h .tana h(l+tana)
(b) ---- h(l-tana)
(c) ---- (d) none ofthese
(1-tana) (1 + tana)
 PRACTICE QUESTIONS
CLASS X: CHAPTER - 9
SOME APPLICATIONS TO TRIGONOMETRY

1. A ve1iical stick 10 cm long casts a shadow 8 cm long. At the same time, a tower casts a shadow
30 m long. Detennine the height ofthe tower.

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

3. A person standing on the bank of a river observes that the angle subtended by a tree on the
opposite bank is 60°. When he retreats 20m from the bank, he finds the angle to be 30°. Find the
height ofthe tree and the breadth ofthe 1iver.

4. A boy is standing on ground and flying a kite with 150m ofstiing at an elevation of30°. Another
boy is standing on the roof of a 25m high building and flying a kite at an elevation of 45° . Find
the length of stting required by the second boy so that the two kites just meet, if both the boys
are on opposite side ofthe kites.

5. An aeroplane flying horizontally 1000m above the grmmd, is observed at an angle of elevation
60° from a point on the grmmd. After a flight of 10 seconds, the angle of elevation at the point of
observation changes to 30°. Find the speed ofthe plane in mis.

6. An aeroplane when flying at a height of 4000 m from the ground passes ve1tically above another
aeroplane at an instant when the angles of the elevation ofthe two planes from the same point on
the ground are 60° and 45° respectively. Find the vertical distance between the aeroplanes at that
instant.

7. An aeroplane at an altitude of 200 m obse1ves the angles of depression ofopposite points on the
two banks ofa 1iver to be 45° and 60°. Find the width ofthe 1iver.

8. The shadow ofa flag staff is three times as long as the shadow of the flag staff when the sun rays
meet the ground at an angle of 600. Find the angle between the sun rays and the ground at the
time oflonger shadow.

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

10. A man in a boat rowing away from lighthouse 100 m high takes 2 minutes to changes the angle
ofelevation ofthe top ofligl1thouse from 60° to 45°. Find the speed ofthe boat.

11. As obse1ved from the top ofa light house, 1 00m above sea level, the angle ofdepression of ship,
sailing directly towards it, changes from 30° to 45°. Detennine the distance travelled by the ship
dming the period ofobse1vation.

12. A man standing on the deck of ship, which is 10m above the water level, obse1ves 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 ofthe hill :from the ship and the heigl1t ofthe hill.
13. The angles of elevation of the top of a tower from two points at a distance of 'a' m and 'b' m
from the base of the tower and in the same straight line with it are complementaiy, then prove
that the height of the tower is ✓ah

14. A tower stands vertically 011 the ground. From a point on the ground, which is 15 m away from
the foot of the tower, the angle of elevatio11 of the top of the tower is found to be 60°. Find the
height of the tower.
15. An electrician has to repair an electric fault 011 a pole of height 5 m. She needs to reach a point
1.3m below the top of the pole to unde1iake the repair work. What should be the length of the
ladder that she should use which, when inclined at an angle of 60° to the horizontal would
enable her to reach the required position? Also,3 how fat· from the foot of the pole should she
place the foot of the ladder? (You may take ✓ = 1. 73)

16. An obse1ver 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the
chimney from her eyes is 45°. What is the height of the chimney?
17. From a point P on the ground the angle of elevation of the top 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 flagstaff from
P is 45°. Find
3 the length of the flagstaff and the distance of the building from the point P. (You
may take ✓ = 1.73)

18. The shadow of a tower standing on a level grom1d is found to be 40 m longer whe11 the Sun's
altitude is 30° than when it is 60°. Find the height of the tower.

19. The angles of depression of the top a11d the bottom of an 8 m tall building from the top of a
multi-storeyed building are 30° and 45°, respectively. Find the height of the multi-storeyed
building and the distance between the two buildings.

20. From a point on a bridge across a river, the angles of depression of the banks on opposite sides of
the river ai·e 30° atid 45°, respectively. If the bridge is at a height of 3 m from the banks, find the
width of the river.

21. 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 towai·ds the
building. Find the distance he walked towai·ds the building.

22. 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.

23. A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of
elevation of the top of the statue is 60° a.11d from the same point the angle of elevation of the top
of the pedestal is 45°. Find the height of the pedestal.

24. The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of
elevatio11 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.

25. A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m
from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is
60°. After some time, the angle of elevation reduces to 30°. Find the distance travelled by the
balloon during the inte1val.
26. A straight highway leads to the foot ofa tower. A man standing at the top ofthe tower observes a
car at an angle of depression of 30°, which is approaching the foot of the tower with a unif01m
speed. Six seconds later, the angle of depression ofthe car is found to be 60° . Find the time taken
by the car to reach the foot ofthe tower from this point.

27. A man on cliff observes a boat an angle ofdepression of 30° which is approaching the shore to
the point immediately beneath the observer with a unifo1m speed. Six minutes later, the angle of
depression ofthe boat is fom1d to be 60°. Find the time taken by the boat to reach the shore.

28. The angles of elevation of the top ofa tower from two points at a distance of4 m and 9 m from
the base of the tower and in the same straight line with it are complementaiy. Prove that the
heigl1t ofthe tower is 6 m

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

30. A tree is broken by the stonn. The top ofthe tree touches the ground making an angle 30° and at
a distance of30 m from the root. Find the height ofthe tree.

31. A tree 12m high, is broken by the storm The top ofthe tree touches the ground making an angle
60°. At what height from the bottom the tree is broken by the stonn.

32. At a point on level ground, the angle ofelevation ofa ve11ical tower is found to be such that its
tangent is .2... In walking 192 m towards the tower, the tangent of the angle of elevation is 2_.
12 4
Find the height ofthe tower.

33. The pilot of an aircraft flying horizontally at a speed of 1200km/hr, observes that the angle of
depression of a point on the ground changes from 30° to 45° in 15 seconds. Find the height at
which the aircraft is flying.

34. If the angle of elevation of the cloud from a point h m above a lake is A and the angle of
. . . . . h(tanB+tanA)
depress1on ofits reflechon
• rn ght ofthe cloud IS ------
• the lake is B, prove that the heio
(tan B - tan A)
35. The angle of elevation of cloud from a point 120m above a lake is 30° and the angle of
depression ofthe reflection ofthe cloud in the lake is 60°. Find the height ofthe cloud.

36. The angle ofelevation of cloud :from a point 60m above a lake is 30° and the angle ofdepression
ofthe reflection ofthe cloud in the lake is 60°. Find the height ofthe cloud.

37. The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 15
seconds, the angle of elevation changes to 30°. If the jet plane is flying at a constant height of
l 500,J3 m, find the speed ofthe jet plane.

38. The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 30
seconds, the angle of elevation changes to 30°. If the jet plane is flying at a constant height of
3600✓3 m, find the speed ofthe jet plane.

39. There ai·e two temples, one on each bank ofriver, just opposite to each other. One temple is 50m
high. From the top ofthis temple, the angles ofdepression ofthe top and foot ofthe other temple
ai·e 30° and 60° respectively. Find the width ofthe river and the height ofother temple.
40. A ladder rests against a wall at an angle a to the horizontal, its foot is pulled away from the wall
through a distant a, so that it slides a distance b down the wall making an angle J3 with the
. a coso.-cosJ3
honzonta1. Show that-=----.
b sinJ3-sino.
41. From a window, h meter above the ground of a house in a street , the angle of elevation and
depression of the top and the foot of another house on the opposite side of the street are 0 and <I>
respectively. Show that the height of the opposite house is h (1 + tan0cot<j>).

42. From a window, 15 meters high above the ground of a house in a street , the angle of elevation
and depression of the top and the foot of another house on the opposite side of the street are 30°
and 45° respectively. Find the height of the opposite house.

43. Two stations due south of a leaning tower which leans towards the n01th are at distances a and b
from its foot. Ifa and J3 are the elevations of the top of the tower from these stations, prove that
. . . . . . b cota -a cotp
its mclinat.ion e to the honzontal is given by cot0 =- - - � - .
b-a

44. The angle of elevation of a cliff from a fixed point is 0 . Afte. r going up a distance of 'k'meters
towards the top of the cliff at an angle of <I> , it is found that the angle of elevation is a. Show that
. liff. k(cos<j>-sin<j>.cota)
the height of the c is-------.
cot0-cota

45. A round balloon of radius r subtends an angle a at the eye of the observer while the angle of
elevation of its centre is J3. Prove that the height of the centre of the balloon is rsinp.cosec�
2
46. The angle of elevation of the top of a tower from a point on the same level as the foot of the
tower is a. On advancing 'p' meters towards the foot of the tower the angle of elevation becomes
tana tanJ3
J3. Show that the height 'h' of the tower is given by h = (p ) m. Also detennine the
tanJ3-tana
height of the tower if p = 150° m, a = 30° and J3 = 60°.

47. From the top of a light- house the angle of depression of two ships on the opposite sides of it are
observed to be a and p. If the height of the light-house be 'h' meter and the line joining the ships
passes through the foot of the light house, show that the distance between the ships is
tana+ tanJ3
h( ) meters.
tana.tanJ3
48. An electrician has to repair on electric fault on a pole of height 4m. she needs to reach a point
1.3m below the top of the pole to undeitake the repair work. What should be the height of the
ladder that she should use at angle of 60° to the horizontal, would enable her reach the required
position? Also, how far the foot of the pole should she place the foot of the ladder.( take ✓3 =
1.732)
49. The angle of elevation of a jet fighter from a point A on the ground is 60°. After a flight of 15
sec, the angle of elevation changes to 300. If the jet is flying at a speed of 720 km/hr, find the
constant height at which the jet is flying.
50. A man on a top of a tower observes a truck at angle of depression a where tana =
}s and sees
that it is moving towards the base of the tower. Ten minutes later, the angle of depression of
tiuck found to be J3 where tanJ3 = ✓5 if the tiuck is moving at unifoim speed detennine how
much more time it will take to reach the base of the tower.
51. At the foot of a mountain the elevation of its summit is 45° ; after ascending 1000m towards the
mountain up a slope of 30° inclination, the elevation is found to be 60°. Find the height of the
mountain.
52. If the angle of elevation of cloud from a point h metres above a lake is a and the angle of
depression of its reflection in the lake be p, prove that the distance of the cloud from the point of
2hseca
observation is- - - -
tanp-tana

53. A ve1iical tower stands on a horizontal plane and is smmounted by a ve1tical flag staff of height
'h'. At a point on the plane, the angles of elevation of the bottom and top of the flag staff are a
htana
and J3 respectively. Prove that the height of the tower is a
tnJ3-tana

54. A man on the top of a ve1iical tower obse1ves a car moving at a unifonn speed coming directly
towards it. If it takes 12 minutes for the angle of depression to change from 30° to 45° , how soon
after this, will the car reach the tower? Give yam answer to the nearest second.

55. Two pillars of equal height and on either side of a road, which is 100m wide. The angles of
depression of the top of the pillars are 60° and 30° at a point on the road between the pillars. Find
the position of the point between the pillars and the height of the tower.

56. The angle of elevation of the top of a tower from a point A due n01th of the tower is a and from
B due west of the tower is J3. If AB= d, show that the height of the tower is d✓sina sin J3
sin 2 a-sin 2 J3

57. The angle of elevation of the top of a tower from a point A due south of the tower is a and from
B due east of the tower is J3. If AB= d, show that the height of the tower is ✓ d
cot 2 a+ cot2 p

58. From an aeroplane ve1tically above a straight horizontal road, the angles of depression of two
consecutive milestones on opposite sides of the aeroplane a1·e obse1ved to be a and J3. Show that
tana tanp
the height in miles of aeroplane above the road is given by .
tana+tanp

59. A tree standing on horizontal plane is leaning towards east. At two points situated at distances a
and b exactly due west on it, angles of elevation of the top are respectively a and J3. Prove that
(b - a) tana tan P .
the height of the top from the ground is
tana-tanp

60. The length of the shadow of a tower standing on level ;lane is found to be 2x metres longer
when the sun's altitude is 30° than when it was 45 . Prove that the height of tower is
x( ✓3 +l)m .

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