# Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 2 - Trigonometry - 1 [Latest edition]

## Chapter 2: Trigonometry - 1

Exercise 2.1Exercise 2.2Miscellaneous Exercise 2
Exercise 2.1 [Pages 21 - 22]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 2 Trigonometry - 1 Exercise 2.1 [Pages 21 - 22]

Exercise 2.1 | Q 1.01 | Page 21

Find the trigonometric function of :

Exercise 2.1 | Q 1.02 | Page 21

Find the trigonometric function of :

30°

Exercise 2.1 | Q 1.03 | Page 21

Find the trigonometric function of :

45°

Exercise 2.1 | Q 1.04 | Page 21

Find the trigonometric function of :

60°

Exercise 2.1 | Q 1.05 | Page 21

Find the trigonometric function of :

150°

Exercise 2.1 | Q 1.06 | Page 21

Find the trigonometric functions of :

180°

Exercise 2.1 | Q 1.07 | Page 21

Find the trigonometric function of :

210°

Exercise 2.1 | Q 1.08 | Page 21

Find the trigonometric function of :

300°

Exercise 2.1 | Q 1.09 | Page 21

Find the trigonometric functions of :

330°

Exercise 2.1 | Q 1.1 | Page 21

Find the trigonometric functions of :

−30°

Exercise 2.1 | Q 1.11 | Page 21

Find the trigonometric functions of :

−45°

Exercise 2.1 | Q 1.12 | Page 21

Find the trigonometric functions of :

−60°

Exercise 2.1 | Q 1.13 | Page 21

Find the trigonometric functions of :

− 90°

Exercise 2.1 | Q 1.14 | Page 21

Find the trigonometric functions of :

−120°

Exercise 2.1 | Q 1.15 | Page 21

Find the trigonometric functions of :

−225°

Exercise 2.1 | Q 1.16 | Page 21

Find the trigonometric functions of :

−240°

Exercise 2.1 | Q 1.17 | Page 21

Find the trigonometric functions of :

−270°

Exercise 2.1 | Q 1.18 | Page 21

Find the trigonometric functions of :

−315°

Exercise 2.1 | Q 2. (i) | Page 21

State the signs of tan 380°

Exercise 2.1 | Q 2. (ii) | Page 21

State the signs of cot 230°

Exercise 2.1 | Q 2. (iii) | Page 21

State the signs of sec 468°

Exercise 2.1 | Q 3 | Page 21

State the signs of cos4c and cos4°. Which of these two functions is greater?

Exercise 2.1 | Q 4. (i) | Page 22

State the quadrant in which θ lies if :

sin θ < 0 and tan θ > 0

Exercise 2.1 | Q 4. (ii) | Page 22

State the quadrant in which θ lies if :

cos θ < 0 and tan θ > 0

Exercise 2.1 | Q 5. (i) | Page 22

Evaluate the following:

sin 30° + cos 45° + tan 180°

Exercise 2.1 | Q 5. (ii) | Page 22

Evaluate the following :

cosec 45° + cot 45° + tan 0°

Exercise 2.1 | Q 5. (iii) | Page 22

Evaluate the following :

sin 30° × cos 45° × tan 360°

Exercise 2.1 | Q 6 | Page 22

Find all trigonometric functions of angle in standard position whose terminal arm passes through point (3, −4).

Exercise 2.1 | Q 7 | Page 22

If cos θ = 12/13, 0 < θ < pi/2, find the value of (sin^2theta - cos^2theta)/(2sinthetacostheta), 1/(tan^2theta)

Exercise 2.1 | Q 8. (i) | Page 22

Using tables evaluate the following :

4 cot 45° – sec2 60° + sin 30°

Exercise 2.1 | Q 8. (ii) | Page 22

Using tables evaluate the following :

cos^2 0 + cos^2  pi/6 + cos^2  pi/3 + cos^2  pi/2

Exercise 2.1 | Q 9. (i) | Page 22

Find the other trigonometric functions:

If cos θ = -3/5 and 180° < θ < 270°.

Exercise 2.1 | Q 9. (ii) | Page 22

Find the other trigonometric functions:

If sec A = -25/7 and A lies in the second quadrant.

Exercise 2.1 | Q 9. (iii) | Page 22

Find the other trigonometric functions:

If cot x = 3/4, x lies in the third quadrant.

Exercise 2.1 | Q 9. (iv) | Page 22

Find the other trigonometric functions:

If tan x = (-5)/12, x lies in the fourth quadrant.

Exercise 2.2 [Page 31]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 2 Trigonometry - 1 Exercise 2.2 [Page 31]

Exercise 2.2 | Q 1 | Page 31

If 2 sinA = 1 = sqrt(2) cosB and pi/2 < A < pi, (3pi)/2 < B < 2pi, then find the value of (tan"A" + tan"B")/(cos"A" - cos"B")

Exercise 2.2 | Q 2 | Page 31

If sin"A"/3 = sin"B"/4 = 1/5 and A, B are angles in the second quadrant then prove that 4cosA + 3cosB = – 5.

Exercise 2.2 | Q 3 | Page 31

If tanθ = 1/2, evaluate (2sin theta + 3cos theta)/(4cos theta + 3sin theta)

Exercise 2.2 | Q 4. (i) | Page 31

Eliminate θ from the following :

x = 3secθ , y = 4tanθ

Exercise 2.2 | Q 4. (ii) | Page 31

Eliminate θ from the following :

x = 6cosecθ , y = 8cotθ

Exercise 2.2 | Q 4. (iii) | Page 31

Eliminate θ from the following :

x = 4cosθ − 5sinθ, y = 4sinθ + 5cosθ

Exercise 2.2 | Q 4. (iv) | Page 31

Eliminate θ from the following :

x = 5 + 6cosecθ, y = 3 + 8cotθ

Exercise 2.2 | Q 4. (v) | Page 31

Eliminate θ from the following:

2x = 3 − 4tanθ, 3y = 5 + 3secθ

Exercise 2.2 | Q 5 | Page 31

If 2sin2θ + 3sinθ = 0, find the permissible values of cosθ.

Exercise 2.2 | Q 6 | Page 31

If 2cos2θ − 11cosθ + 5 = 0 then find possible values of cosθ.

Exercise 2.2 | Q 7 | Page 31

Find the acute angle θ such that 2 cos2θ = 3 sin θ.

Exercise 2.2 | Q 8 | Page 31

Find the acute angle θ such that 5tan2θ + 3 = 9secθ.

Exercise 2.2 | Q 9 | Page 31

Find sinθ such that 3cosθ + 4sinθ = 4

Exercise 2.2 | Q 10 | Page 31

If cosecθ + cotθ = 5, then evaluate secθ.

Exercise 2.2 | Q 11 | Page 31

If cotθ = 3/4 and π < θ < (3pi)/4 then find the value of 4cosecθ + 5cosθ.

Exercise 2.2 | Q 12. (i) | Page 31

Find the Cartesian co-ordinates of points whose polar coordinates are :

(3, 90°)

Exercise 2.2 | Q 12. (ii) | Page 31

Find the Cartesian co-ordinates of points whose polar coordinates are :

(1,180°)

Exercise 2.2 | Q 13. (i) | Page 31

Find the polar coordinates of points whose cartesian coordinates are :

(5, 5)

Exercise 2.2 | Q 13. (ii) | Page 31

Find the polar coordinates of points whose cartesian coordinates are :

(1, sqrt(3))

Exercise 2.2 | Q 13. (iii) | Page 31

Find the polar co-ordinates of points whose Cartesian co-ordinates are:

(–1, –1)

Exercise 2.2 | Q 13. (iv) | Page 31

Find the polar co-ordinates of points whose Cartesian co-ordinates are:

(- sqrt(3), 1)

Exercise 2.2 | Q 14. (i) | Page 31

Find the value of :

sin  (19pi^"c")/3

Exercise 2.2 | Q 14. (ii) | Page 31

Find the value of :

cos 1140°

Exercise 2.2 | Q 14. (iii) | Page 31

Find the values of:

cot  (25pi^"c")/3

Exercise 2.2 | Q 15. (i) | Page 31

Prove the following identities:

(1 + tan^2 "A") + (1 + 1/tan^2"A") = 1/(sin^2 "A" - sin^4"A")

Exercise 2.2 | Q 15. (ii) | Page 31

Prove the following identities:

(cos2A – 1) (cot2A + 1) = −1

Exercise 2.2 | Q 15. (iii) | Page 31

Prove the following identities:

(sinθ + sec θ)2 + (cosθ + cosec θ)2 = (1 + cosecθ sec θ)2

Exercise 2.2 | Q 15. (iv) | Page 31

Prove the following identities:

(1 + cot θ – cosec θ)(1 + tan θ + sec θ) = 2

Exercise 2.2 | Q 15. (v) | Page 31

Prove the following identities:

tan^3theta/(1 + tan^2theta) + cot^3theta/(1 + cot^3theta = secθ cosecθ – 2sinθ cosθ

Exercise 2.2 | Q 15. (iv) | Page 31

Prove the following identities:

1/(sectheta + tantheta) - 1/costheta = 1/costheta - 1/(sectheta - tantheta)

Exercise 2.2 | Q 15. (vii) | Page 31

Prove the following identities:

sintheta/(1 + costheta) + (1 + costheta)/sintheta = 2cosecθ

Exercise 2.2 | Q 15. (viii) | Page 31

Prove the following identities:

tantheta/(sectheta - 1) = (sectheta + 1)/tantheta

Exercise 2.2 | Q 15. (ix) | Page 31

Prove the following identities:

cottheta/("cosec"  theta - 1) = ("cosec"  theta + 1)/cot theta

Exercise 2.2 | Q 15. (x) | Page 31

Prove the following identities:

(secA + cosA)(secA − cosA) = tan2A + sin2A

Exercise 2.2 | Q 15. (xi) | Page 31

Prove the following identities:

1 + 3cosec2θ cot2θ + cot6θ = cosec6θ

Exercise 2.2 | Q 15. (xii) | Page 31

Prove the following identities:

(1 - sec theta + tan theta)/(1 + sec theta - tan theta) = (sec theta + tan theta - 1)/(sec theta + tan theta + 1)

Miscellaneous Exercise 2 [Pages 32 - 34]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 2 Trigonometry - 1 Miscellaneous Exercise 2 [Pages 32 - 34]

Miscellaneous Exercise 2 | Q I. (1) | Page 32

Select the correct option from the given alternatives:

The value of the expression cos1°. cos2°. cos3° … cos179° =

• – 1

• 0

• 1/sqrt(2)

• 1

Miscellaneous Exercise 2 | Q I. (2) | Page 32

Select the correct option from the given alternatives:

tan"A"/(1 + sec"A") + (1 + sec"A")/tan"A" is equal to

• 2cosecA

• 2secA

• 2sinA

• 2cosA

Miscellaneous Exercise 2 | Q I. (3) | Page 32

Select the correct option from the given alternatives:

If α is a root of 25cos2θ + 5cosθ – 12 = 0, pi/2 < α < π, then sin2α is equal to

• -24/25

• -13/18

• 13/18

• 24/25

Miscellaneous Exercise 2 | Q I. (4) | Page 33

Select the correct option from the given alternatives:

If θ = 60°, then (1 + tan^2theta)/(2tantheta) is equal to

• sqrt(3)/2

• 2/sqrt(3)

• 1/sqrt(3)

• sqrt(3)

Miscellaneous Exercise 2 | Q I. (5) | Page 33

Select the correct option from the given alternatives:

If secθ = m and tanθ = n, then 1/"m"{("m + n") + 1/(("m + n"))} is equal to

• 2

• mn

• 2m

• 2n

Miscellaneous Exercise 2 | Q I. (6) | Page 33

Select the correct option from the given alternatives:

If cosecθ + cotθ = 5/2, then the value of tanθ is

• 14/25

• 20/21

• 21/20

• 15/16

Miscellaneous Exercise 2 | Q I. (7) | Page 33

Select the correct option from the given alternatives:

1 - sin^2theta/(1 + costheta) + (1 + costheta)/sintheta - sintheta/(1 - costheta) equals

• 0

• 1

• sin θ

• cos θ

Miscellaneous Exercise 2 | Q I. (8) | Page 33

Select the correct option from the given alternatives:

If cosecθ − cotθ = q, then the value of cot θ is

• (2"q")/(1 + "q"^2)

• (2"q")/(1 - "q"^2)

• (1 - "q"^2)/(2"q")

• (1 + "q"^2)/(2"q")

Miscellaneous Exercise 2 | Q I. (9) | Page 33

Select the correct option from the given alternatives:

The cotangent of the angles pi/3, pi/4 and pi/6 are in

• A.P.

• G.P.

• H.P.

• Not in progression

Miscellaneous Exercise 2 | Q I. (10) | Page 33

Select the correct option from the given alternatives:

The value of tan1°.tan2°tan3°..... tan89° is equal to

• – 1

• 1

• pi/2

• 2

Miscellaneous Exercise 2 | Q II. (1) a. | Page 33

Find the trigonometric functions of :

90°

Miscellaneous Exercise 2 | Q II) 1) b. | Page 33

Find the trigonometric functions of:

120°

Miscellaneous Exercise 2 | Q II. (1) c. | Page 33

Find the trigonometric functions of :

225°

Miscellaneous Exercise 2 | Q II. (1) d. | Page 33

Find the trigonometric functions of :

240°

Miscellaneous Exercise 2 | Q II. (1) e. | Page 33

Find the trigonometric functions of :

270°

Miscellaneous Exercise 2 | Q II. (1) f. | Page 33

Find the trigonometric functions of :

315°

Miscellaneous Exercise 2 | Q II. (1) g. | Page 33

Find the trigonometric functions of :

−120°

Miscellaneous Exercise 2 | Q II. (1) h. | Page 33

Find the trigonometric functions of :

−150°

Miscellaneous Exercise 2 | Q II. (1) i. | Page 33

Find the trigonometric functions of :

−180°

Miscellaneous Exercise 2 | Q II. (1) j. | Page 33

Find the trigonometric functions of :

−210°

Miscellaneous Exercise 2 | Q II. (1) k. | Page 33

Find the trigonometric functions of :

−300°

Miscellaneous Exercise 2 | Q II. (1) l. | Page 33

Find the trigonometric functions of :

−330°

Miscellaneous Exercise 2 | Q II. (2) i) | Page 33

State the signs of cosec 520°

Miscellaneous Exercise 2 | Q II. (2) ii) | Page 33

State the signs of cot 1899°

Miscellaneous Exercise 2 | Q II. (2) iii) | Page 33

State the signs of sin 986°

Miscellaneous Exercise 2 | Q II. (3) i) | Page 33

State the quadrant in which θ lies if tan θ < 0 and sec θ > 0

Miscellaneous Exercise 2 | Q II. (3) ii) | Page 33

State the quadrant in which θ lies if sin θ < 0 and cos θ < 0

Miscellaneous Exercise 2 | Q II. (3) iii) | Page 33

State the quadrant in which θ lies if sin θ > 0 and tan θ < 0

Miscellaneous Exercise 2 | Q 4 | Page 33

Which is greater sin(1856°) or sin(2006°)?

Miscellaneous Exercise 2 | Q 5 | Page 33

Which of the following is positive? sin(−310°) or sin(310°)

Miscellaneous Exercise 2 | Q 6 | Page 33

Show that 1 − 2sinθ cosθ ≥ 0 for all θ ∈ R.

Miscellaneous Exercise 2 | Q 7 | Page 33

Show that tan2θ + cot2θ ≥ 2 for all θ ∈ R

Miscellaneous Exercise 2 | Q 8 | Page 33

If sinθ = (x^2 - y^2)/(x^2 + y^2) then find the values of cosθ, tanθ in terms of x and y.

Miscellaneous Exercise 2 | Q 9 | Page 33

If sec θ = sqrt(2) and (3pi)/2 < theta < 2pi then evaluate (1 + tantheta + "cosec"theta)/(1 + cottheta - "cosec"theta)

Miscellaneous Exercise 2 | Q 10. (i) | Page 33

Prove the following:

sin2A cos2B + cos2A sin2B + cos2A cos2B + sin2A sin2B = 1

Miscellaneous Exercise 2 | Q 10. (ii) | Page 33

Prove the following:

((1 + cot theta + tan theta)(sin theta - costheta)) /(sec^3theta - "cosec"^3theta)= sin2θ cos2θ

Miscellaneous Exercise 2 | Q 10. (iii) | Page 33

Prove the following:

(tan theta + 1/costheta)^2 + (tan theta - 1/costheta)^2 = 2((1 + sin^2theta)/(1 - sin^2theta))

Miscellaneous Exercise 2 | Q 10. (iv) | Page 33

Prove the following:

2 sec2θ – sec4θ – 2cosec2θ + cosec4θ = cot4θ – tan4θ

Miscellaneous Exercise 2 | Q 10. (v) | Page 33

Prove the following:

sin4θ + cos4θ = 1 – 2 sin2θ cos2θ

Miscellaneous Exercise 2 | Q 10. (vi) | Page 33

Prove the following:

2(sin6θ + cos6θ) – 3(sin4θ + cos4θ) + 1 = 0

Miscellaneous Exercise 2 | Q 10. (vii) | Page 34

Prove the following:

cos4θ − sin4θ +1= 2cos2θ

Miscellaneous Exercise 2 | Q 10. (viii) | Page 34

Prove the following:

sin4θ +2sin2θ . cos2θ = 1 − cos4θ

Miscellaneous Exercise 2 | Q 10. (ix) | Page 33

Prove the following:

(sin^3theta + cos^3theta)/(sintheta + costheta) + (sin^3theta - cos^3theta)/(sintheta - costheta) = 2

Miscellaneous Exercise 2 | Q 10. (x) | Page 34

Prove the following:

tan2θ − sin2θ = sin4θ sec2θ

Miscellaneous Exercise 2 | Q 10. (xi) | Page 34

Prove the following:

(sinθ + cosecθ)2 + (cosθ + secθ)2 = tan2θ + cot2θ + 7

Miscellaneous Exercise 2 | Q 10. (xii) | Page 34

Prove the following:

sin8θ − cos8θ = (sin2θ − cos2θ) (1 − 2 sin2θ cos2θ)

Miscellaneous Exercise 2 | Q 10. (xiii) | Page 34

Prove the following:

sin6A + cos6A = 1 − 3sin2A + 3 sin4A

Miscellaneous Exercise 2 | Q 10. (xiv) | Page 34

Prove the following:

(1 + tanA · tanB)2 + (tanA − tanB)2 = sec2A · sec2B

Miscellaneous Exercise 2 | Q 10. (xv) | Page 34

Prove the following:

(1 + cot  +  "cosec" theta)/(1 - cot  +  "cosec" theta) = ("cosec" theta  + cottheta - 1)/(cottheta - "cosec"theta + 1)

Miscellaneous Exercise 2 | Q 10. (xvi) | Page 34

Prove the following:

(tantheta + sectheta - 1)/(tantheta + sectheta + 1) = tantheta/(sec theta + 1)

Miscellaneous Exercise 2 | Q 10. (xvii) | Page 34

Prove the following:

("cosec"theta + cottheta - 1)/( "cosec"theta + cot theta + 1) =(1-sintheta)/costheta

Miscellaneous Exercise 2 | Q 10. (xviii) | Page 34

Prove the following:

("cosec"theta + cottheta + 1)/(cottheta + "cosec" theta - 1) = cottheta/("cosec"theta - 1)

## Chapter 2: Trigonometry - 1

Exercise 2.1Exercise 2.2Miscellaneous Exercise 2

## Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 2 - Trigonometry - 1

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 2 (Trigonometry - 1) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 2 Trigonometry - 1 are Introduction of Trigonometry, Trigonometric Functions with the Help of a Circle, Signs of Trigonometric Functions in Different Quadrants, Range of Cosθ and Sinθ, Trigonometric Functions of Specific Angles, Trigonometric Functions of Negative Angles, Fundamental Identities, Periodicity of Trigonometric Functions, Domain and Range of Trigonometric Functions, Graphs of Trigonometric Functions, Polar Co-ordinate System.

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