10th_maths_hots_chapter-6-trignometry - Weebly

JSUNIL TUTORIAL PANJABI COLONY GALI O1 UNIT-6 TRIGONOMETRY "The mathematician is fascinated with the marvelous beauty of the forms he constructs, and in their beauty he finds everlasting truth."

1.

If xcosθ – ysinθ = a, xsinθ + ycos θ = b, prove that x2+y2=a2+b2.

Ans: xcosθ - y sinθ = a xsinθ + y cosθ = b Squaring and adding x2+y2=a2+b2. 2.

Prove that sec2θ+cosec2θ can never be less than 2.

Ans: S.T Sec2θ + Cosec2θ can never be less than 2. If possible let it be less than 2. 1 + Tan2θ + 1 + Cot2θ < 2. 2 + Tan2θ + Cot2θ (Tanθ + Cotθ)2 < 2. Which is not possible. 3.

, show that 3cos -4cos3 = 0.

If sin =

Ans: Sin

=½ = 30o Substituting in place of

4.

=30o. We get 0.

If 7sin2 +3cos2 = 4, show that tan =

.

1

Ans: If 7 Sin2 + 3 Cos2 = 4 S.T. Tan

3

7 Sin

2

2

+ 3 Cos

= 4 (Sin

2

2

+ Cos )

3 Sin2 = Cos2 Sin

2

Cos

2

=

1 3

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JSUNIL TUTORIAL PANJABI COLONY GALI O1 Tan2 =

1 3

1

Tan =

3

5.

If cos +sin =

cos , prove that cos - sin =

Ans: Cos + Sin = 2 Cos ( Cos + Sin )2 = 2Cos2 Cos2 + Sin2 +2Cos Sin = 2Cos2 Cos2 - 2Cos Sin + Sin2 = 2Sin2 (Cos - Sin )2 = 2Sin2 or Cos - Sin = 2 Sin . 6.

sin .

2Sin2 = 2 - 2Cos2 1- Cos2 = Sin2 & 1 - Sin2 = Cos2

If tanA+sinA=m and tanA-sinA=n, show that m2-n2 = 4

Ans: TanA + SinA = m TanA – SinA = n. 2 2 m -n =4 mn . m2-n2= (TanA + SinA)2-(TanA - SinA)2 = 4 TanA SinA RHS 4 mn = 4 TanA SinA (TanA SinA ) =4

2

Tan A 2

=4

Sin A

2

Sin A 2

Sin ACos Cos

2

2

A

A

4

=4

Sin A Cos

2

A

2

=4

Sin A Cos

2

= 4 TanA SinA

A

m2 – n2 = 4 mn

7.

If secA=

, prove that secA+tanA=2x or

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.

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JSUNIL TUTORIAL PANJABI COLONY GALI O1 1

Ans: Sec = x +

4x 1

Sec2 =( x +

4x 1

Tan2 = ( x +

)2 )2-1

4x 1 2 ) 4x

Tan2 = ( x -

1

Tan = + x -

4x

Sec + Tan = x +

1 4x

+ x-

1 4x

1

= 2x or 8.

(Sec2 = 1 + Tan2 )

2x

If A, B are acute angles and sinA= cosB, then find the value of A+B.

Ans: A + B = 90o

9.

a)Solve for , if tan5 = 1.

Ans: Tan 5 = 1

=

b)Solve for Sin 1

1

Cos

Sin

2

Sin

Sin

(1

Cos

1

Sin

Sin

2 Cos (1

4.

Sin

4

Cos

2

4

)

Cos

Sin

2

Cos

Sin 1( Cos )

2

1

Cos

Cos

=9o.

5

Sin

if 1

Ans:

45

2

2 Cos

4

Cos

4 )

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JSUNIL TUTORIAL PANJABI COLONY GALI O1 2

(1

Cos

)

Sin

(1

Cos

)

2

4

4

Sin 1

Sin =

2

Sin = Sin30 = 30o

10.

Ans:

If Cos

Cos

m

Cos

n

Sin

m2=

Cos

2

Cos

2

n2=

Cos

2

Sin

2

LHS = (m2+n2) Cos2 Cos

2

Cos

2

= Cos

2

Cos

2

Sin

2

=

Cos

2

Sin

2

1 Cos

2

Sin

2

2

Cos

2

=n2

(m2+n2) Cos

11.

Cos

2

=n2

If 7 cosec -3cot = 7, prove that 7cot - 3cosec = 3.

Ans: 7 Cosec -2Cot =7 P.T 7Cot - 3 Cosec =3 7 Cosec -3Cot =7 7Cosec -7=3Cot 7(Cosec -1)=3Cot Page downloaded from cbseguide.weebly.com

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JSUNIL TUTORIAL PANJABI COLONY GALI O1 7(Cosec -1) (Cosec +1)=3Cot (Cosec +1) 7(Cosec2 -1)=3Cot (Cosec +1) 7Cot2 =3 Cot (Cosec +1) 7Cot = 3(Cosec +1) 7Cot -3 Cosec =3 12.

2(sin 6 +cos6 ) – 3(sin4 +cos4 )+1 = 0

Ans: (Sin2 )3 + (Cos2 )3-3 (Sin4 +(Cos4 )+1=0 Consider (Sin2 )3 +(Cos2 )3 (Sin2 +Cos2 )3-3 Sin2 Cos2 ( Sin2 +Cos2 ) = 1- 3Sin2 Cos2 Sin4 +Cos4 (Sin2 )2+(Cos2 )2 = (Sin2 +Cos2 )2-2 Sin2 Cos2 = 1- 2 Sin2 Cos2 = 2(Sin6 +Cos6 )-3(Sin4 +Cos4 ) +1 = 2 (1-3 Sin2 Cos2 )-3 (1-2 Sin2 +Cos2 )+1 13.

5(sin8A- cos8A) = (2sin2A – 1) (1- 2sin2A cos2 A)

Ans: Proceed as in Question No.12 14.

5

If tan =

Ans: Tan = 5 15.

+ =90o what is the value of cot .

&

6

i.e.

6

Cot = 5

6

Since

+

= 90o.

What is the value of tan in terms of sin .

Ans: Tan

=

Sin Cos

Tan

Sin

= 1

16.

Sin

2

If Sec +Tan =4 find sin , cos

Ans: Sec

+ Tan

1

Sin

Cos

Cos

1

Sin

=4 =4

4

Cos

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JSUNIL TUTORIAL PANJABI COLONY GALI O1 (1

Sin

)

2

16

2

Cos

apply (C & D) =

(1

Sin

(1

Sin

2 (1

)

2

)

2

2

Cos

2

Sin )

2 Sin (1

1

Cos

=

Sin

Sin =

=

16

1

16

1

17

=

15

Sin )

17 15

15 17

Cos = 1 Sin 2 1

2

15

=

17

17.

8 17

p

2

1

p

2

1

P

2

1

P

2

1

Sec +Tan =p, prove that sin =

Ans: Sec + Tan = P.

P.T Sin =

Proceed as in Question No.15 18.

Prove geometrically the value of Sin 60o

Ans:

Exercise for practice.

19.

If

1

tan

3

1

1

tan

3

1

Ans:

Exercise for practice.

20.

If 2x=sec and

Ans:

2 x

,show that

sin cos 2

=1

= tan ,then find the value of 2 x 2

1 x

2

.

(Ans:1)

Exercise for practice.

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