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some basic questions to enhance understanding of trignometry

### Transcript of trignometry basic questions

• L EC TURE

22 Trigonometric Functions

1. TABLE REPRESENTING ALL TRIGONAL RATIOS (TRs) IN TERMS OF ONE TRIGONAL RATIO

sin cos tan cot sec cosec

sin sin 21 cos 2tan

1 tan

+ 21

1 cot+ 2sec 1

sec

1cosec

cos 21 sin cos

2

11 tan+ 2

cot1 cot

+

1sec

2cosec 1cosec

tan 2sin

1 sin

21 cos

cos

tan

1cot 2sec 1 2

1cosec 1

cot21 sin

sin

2cos

1 cos

1tan cot 2

1sec 1

2cosec 1

sec 21

1 sin 1

cos 21 tan+ 21 cot

cot+

sec 2

coseccosec 1

1. TABLE REPRESENTING ALL TRIGONAL RATIOS (TR

2. T-RATIOS (OR TRIGONOMETRIC FUNCTIONS)

sin ,cos , tan ,p b p

h h b = = =

cosec ,sec ,cot ;h h b

p b p = = =

2. T-RATIOS (OR TRIGONOMETRIC

O

p

B

A

h

bp + b = h2 2 2

p perpendicular, b base and h stands for hypotenuse.

• A.18 Trigonometric Functions

3. FOLLOWING ARE SOME OF THE FUNDAMENTAL TRIGONOMETRIC IDENTITIES

(i) 1sincosec

=

or 1cosecsin

=

(ii) 1cossec

=

or 1seccos

=

(iii) 1cottan

=

or 1tancot

=

(iv) sintancos

=

or coscotsin

=

(v) sin2+cos2=1orsin2 =1cos2orcos2 =1sin2

(vi) sec2tan2=1 or1+tan2sin2

or 1sec tansec tan

=+

(vii) cosec2cot2=1or1+cot2=cosec2

or 1cosec cotcosec cot

=+

4. MAXIMUM AND MINIMUM VALUES OF TRIGONOMETRICAL FUNCTIONS

(i) 1 sin x 1 (ii) 1 cos x 1 (iii) < tan x < (iv) < cot x < (v) |sec x | 1 i.e., sec x 1orsecx 1 (vi) |cosec x| 1 i.e., cosec x 1orcosecx 1

3. FOLLOWING ARE SOME OF THE

4. MAXIMUM AND MINIMUM VALUES OF

(vii) sin2 x +cosec2 x 2, x R (viii) cos2 x +sec2 x 2, x R (ix) tan2 x +cot2 x 2, x R (x) |sin x+cosecx | 2 (xi) |cos x+secx | 2 (xii) |tan x+cotx | 2 (xiii) If a sin x+b cos x=c, then (a cos xb sin x)2

=a2 + b2 c2

5. SOME USEFUL RESULTS

(i) sin2+cos4=sin4+cos2=1sin2cos2 (ii) sin4+cos4=12sin2cos2 (iii) sin6+cos6 =13sin2cos2 (iv) sin4cos4 =12cos2 (v) sin8cos8=(sin2cos2)(12sin2

cos2) (vi) sec2+cosec2 =sec2cosec2=tan2+

cot2+2

(vii) 1 1 sinsec tansec tan 1 sin

+ = + =

(viii) 1 1 sinsec tansec tan 1 sin

= =+ +

(ix) 1 1 coscosec cotcosec cot 1 cos

+ = + =

(x) 1 1 coscosec cotcosec cot 1 cos

= =+ +

(xi) 1 sin cos sec tancos 1 sin = =

+

(xii) 1 cos sin cosec cotsin 1 cos = =

+

(xiii) 1 sin cos sec tancos 1 sin = =

+

(xiv) 1 cos sin cosec cotsin 1 cos = =

+

5. SOME USEFUL RESULTS

(i) sin

• Trigonometric Functions A.19

1. Show that 3 3

(1 tan cot )(sin cos )sec cosec

+ +

Solution

L.H.S.=3 3

3 3

sin cos1 (sin cos )(1 tan cot )(sin cos ) cos sin

1 1sec coseccos sin

+ + + + =

3 3

3 3

sin cos1 (sin cos )(1 tan cot )(sin cos ) cos sin

1 1sec coseccos sin

+ + + + =

2 2

3 3

3 3

(sin cos sin cos )(sin cos )sin cossin cossin cos

+ + =

3 3 3 3

3 3

(sin cos ) sin cossin cos (sin cos )

= 2 2 3 3( ( )( ) )a b ab a b a b+ + =

2 2sin cos= = R.H.S.

2. If cos2sin2=tan2,thenshowthattan2 =cos2sin2.

Solution

Given, cos2sin2=tan22 2 2

2 2 2

cos sin sincos sin cos

= +

2 2 2

2 2 2

cos sin coscos sin sin

+ =

2 2( cos sin 1) + =Applying componendo and dividendo, we get

2 2 2 2 2 2

2 2 2 2 2 2

cos sin cos sin cos sincos sin cos sin cos sin

+ + + = + +

2

2 2 2

2cos 12sin cos sin

=

22 2

2

sin cos sincos

=

2 2 2tan cos sin , = as desired.

3. If sinsin

A mB= and cos

cosA nB= ,findthevalueof

tan B; n2 < 1 < m2.

Solution

Given sin sin sinsin

A m A m BB= = .........(1)

and cos cos coscos

A n A n BB= = ..........(2)

By squaring (1) and (2) and adding, we get

2 2 2 21 sin cosm B n B= +2

2 22 2

1 sincos cos

Bm nB B

= + (Dividing by cos2 B)

2 2 2 2sec tan = +B m B n

2 2 2 21 tan tan + = +B m B n 2

2 2 22

11 ( 1) tan tan1

nn m B Bm

2 = =

2

2

1tan1

nBm =

4. Eliminatebetween cosec sin , = a sec cos = b.

Solution

Given cosec sin a = .............(1)and sec cos b = .............(2)

From (1), 21 cossin

sin sina a = =

.............(3)

From (2), 21 sincos

cos cosb b = =

..............(4)Squaring(3)andmultiplyingby(4),weget

3 2 2 1/3cos cos ( )a b a b = = ............(5)

Squaring(4)andmultiplyingby(3),weget3 2 2 1/3sin sin ( )ab ab = = ............(6)

• A.20 Trigonometric Functions

5. Eliminateand between cos cos , cos sin , sin .x r y r z r= = =

Solution

Given cos cos , cos sinx r y r= = andsin .= z r

Squaring and adding, we get2 2 2+ +x y z

2 2 2 2 2 2 2 2cos cos cos sin sinr r r= + + 2 2 2 2 2 2cos (cos sin ) sinr r= + + 2 2 2 2(sin cos )r r= + =

2 2[ sin cos 1] + =

6. If sin cosandsin cos

m n = =

, than prove that

22 2

2

1tan ; 1 .1

m n n mn m

= < .

[IIT-JEE-1981]

Solution

sin cos 22+ =

• A.26 Trigonometric Functions

1. Ifsin= 22

1tt+

,thencosisequalto

(a) 22

1tt

(b) 22

1tt+

(c) 2

2

11

tt

+

(d) 2

2

11

tt

+

2. sin cos1 cot 1 tan

+

is equal to

[Karnataka CET-1998]

(a) 0 (b) 1(c) cossin (d) cos+sin

3. If for real values of x,cos=x+ 1x

, then

[MPPET-1996](a) isanacuteangle(b) isarightangle(c) isanobtuseangle(d) Novaluesofispossible

4. The equation sec2 = 24

( )xy

x y+ is only

possible when [MPPET-1986; IIT-1996](a) x=y (b) x < y(c) x > y (d) None of these

5. Which of the following relations is correct? [WBJEE-91](a) sin 1< sin 1o

(b) sin 1 > sin 1o

(c) sin1=sin1o

(d) sin1 sin1180

=

6. Ifsin+cosec=2,thensin2+cosec2is equal to [MPPET-1992; MNR-1990; UPSEAT-2002](a) 1 (b) 4(c) 2 (d) None of these

7. Iftan= 2021

,coswillbe [MPPET-1994]

(a) 2041

(b) 121

(c) 2129

(d) 20

21

8. If x=a cos3,y=b sin3,then(a) (a/x)2/3+(b/y)2/3=1(b) (b/x)2/3+(a/y)2/3=1(c) (x/a)2/3+(y/b)2/3=1(d) (x/b)2/3+(y/a)2/3=1

9. If x=sec+tan,thenx+ 1x

is equal to

[MPPET-1986](a) 1 (b) 2sec(c) 2 (d) 2tan

10. If(1+sinA)(1+sinB)(1+sinC)=(1sinA) (1sinB)(1sinC), then each side is equal to(a) sin A sin B sin C(b) cosA cosB cos C(c) sin A cos B cos C(d) cos A sin B sin C

11. If sin1+ sin2+ sin3=3, thencos1+cos2+cos3 is equal to [EAMCET-1994](a) 3 (b) 2(c) 1 (d) 0

12. Iftancot=aandsin+cos=b, then (b21) 2 (a2+4)isequalto [WB JEE-1979](a) 2 (b) 4(c) 4 (d) 4

13. Iftan=sin

1 cosx

x

and tan =

sin1 cos

yy

,

then xy

is equal to [MPPET-1991]

(a) sinsin

(b)

sinsin

(c) sin1 cos

(d) sin1 cos

• Trigonometric Functions A.27

14. Ifsec+tan=p,thentanisequalto [MPPET-1994]

(a) 22

1p

p (b)

2 12

pp

(c) 2 12

pp+ (d) 2

21

pp +

15. If 2sin cos

and =1 cos sin 1 sin

p q =+ + + , then

[MPPET-2001]

(a) pq=1 (b) 1qp=

(c) qp=1 (d) q+p=1

16. Theminimumvalueof9tan2+4cot2 is(a) 13 (b) 9

(c) 6 (d) 12

17. The maximum value of 4 sin2 x+3cos2 x is [Karnataka CET-2003](a) 3 (b) 4

(c) 5 (d) 7

18. Which value of k, (cosx+sinx) 2 +k sin x cos x1=0isidentity? [Kerala (Engg.)-2001](a) 1 (b) 2

(c) 0 (d) 1

20. If sinx+sin2x=1,thencos8x+2cos6 x+cos4x is equal to(a) 0 (b) 1(c) 2 (d) 1

21. The least value of 2 sin2 + 3 cos2 is [MPPET-2010]

(a) 1 (b) 2 (c)3 (d)5

22. If 1sec ,4

A xx

= + then the value of sec A+tan A is [MPPET-2010](a) 3x (b)

3

x

(c)

2x (d) 2x

1. (c) Given 22sin

1tt

=+

By Pythagoras theoremAC2=AB2+BC2

(1+t2)2=(2t)2+BC2

(1+t2)2=4t2+BC2

BC2=(1+t2)24t2=1+t4+2t24t2=1+t42t2

BC2=(1t2) BC=1t2

2

2

1cos1

BC tAC t

= =+

Second Method

22

2 tan2 2sin .1 1 tan

2

tt

= = + + If tan

2t =

%WW

&

\$

22

22

1 tan 12cos11 tan

2

tt

= = ++

2. (d) Given sin cos1 cot 1 tan

+

sin coscos sin1 1sin cos

= +

sin cossin cos cos sin

sin cos

= +

2 2sin cos

sin cos cos sin +

• A.28 Trigonometric Functions

2 2sin cos

sin cos sin cos

2 2sin cos

sin cos

(sin cos )(sin cos )(sin cos )

+

sin+cos

3. (d)Givenequationiscos=x+1/xor x2xcos+1=0for real value of x By B2 > 4AC

A=1,B=cos,C=1

cos2>4(1)(1)or|cos|2

Whichisimpossiblebecause|cos|