# Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 3 - Trigonometric Functions [Latest edition]

## Chapter 3: Trigonometric Functions

Exercise 3.1Exercise 3.2Exercise 3.3Miscellaneous exercise 3
Exercise 3.1 [Page 75]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 3 Trigonometric Functions Exercise 3.1 [Page 75]

Exercise 3.1 | Q 1.1 | Page 75

Find the principal solution of the following equation:

cosθ = (1)/(2)

Exercise 3.1 | Q 1.2 | Page 75

Find the principal solution of the following equation:

Sec θ = (2)/sqrt(3)

Exercise 3.1 | Q 1.3 | Page 75

Find the principal solution of the following equation :

cotθ = √3

Exercise 3.1 | Q 1.4 | Page 75

Find the principal solution of the following equation:

cotθ = 0

Exercise 3.1 | Q 2.1 | Page 75

Find the principal solution of the following equation:

sin θ = -1/2

Exercise 3.1 | Q 2.2 | Page 75

Find the principal solution of the following equation:

tan θ = – 1

Exercise 3.1 | Q 2.3 | Page 75

Find the principal solution of the following equation:

sqrt(3)cosecθ+ 2 = 0

Exercise 3.1 | Q 3.1 | Page 75

Find the general solution of the following equation:

sinθ = 1/2.

Exercise 3.1 | Q 3.2 | Page 75

Find the general solution of the following equation :

cosθ = sqrt(3)/(2)

Exercise 3.1 | Q 3.3 | Page 75

Find the general solution of the following equation:

tan θ = (1)/(sqrt(3))

Exercise 3.1 | Q 3.4 | Page 75

Find the general solution of the following equation:

cot θ = 0.

Exercise 3.1 | Q 4.1 | Page 75

Find the general solution of the following equation:

sec θ = sqrt(2).

Exercise 3.1 | Q 4.2 | Page 75

Find the general solution of the following equation:

cosec θ = - √2.

Exercise 3.1 | Q 4.3 | Page 75

Find the general solution of the following equation:

tan θ = - 1

Exercise 3.1 | Q 5.1 | Page 75

Find the general solution of the following equation:

sin 2θ = 1/2

Exercise 3.1 | Q 5.2 | Page 75

Find the general solution of the following equation:

tan (2θ)/(3) = √3.

Exercise 3.1 | Q 5.3 | Page 75

Find the general solution of the following equation:

cot 4θ = – 1

Exercise 3.1 | Q 6.1 | Page 75

Find the general solution of the following equation:

4cos2θ  = 3.

Exercise 3.1 | Q 6.2 | Page 75

Find the general solution of the following equation:

4sin2θ = 1.

Exercise 3.1 | Q 6.3 | Page 75

Find the general solution of the following equation:

cos 4θ = cos 2θ

Exercise 3.1 | Q 7.1 | Page 75

Find the general solution of the following equation:

sin θ = tan θ.

Exercise 3.1 | Q 7.2 | Page 75

Find the general solution of the following equation:

tan3θ = 3 tanθ.

Exercise 3.1 | Q 7.3 | Page 75

Find the general solution of the following equation:

cos θ + sin θ = 1.

Exercise 3.1 | Q 8.1 | Page 75

State whether the following equation have solution or not?

cos 2θ = – 1

Exercise 3.1 | Q 8.2 | Page 75

State whether the following equation has a solution or not?

cos2θ = – 1.

Exercise 3.1 | Q 8.3 | Page 75

State whether the following equation has a solution or not?

2sinθ = 3

Exercise 3.1 | Q 8.4 | Page 75

State whether the following equation have solution or not?

3 tanθ = 5

Exercise 3.2 [Page 88]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 3 Trigonometric Functions Exercise 3.2 [Page 88]

Exercise 3.2 | Q 1.1 | Page 88

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

(sqrt(2), pi/4)

Exercise 3.2 | Q 1.2 | Page 88

Find the Cartesian coordinates of the point whose polar coordinates are :

(4,  pi/2)

Exercise 3.2 | Q 1.3 | Page 88

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

(3/4, (3pi)/4)

Exercise 3.2 | Q 1.4 | Page 88

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

(1/2, (7pi)/3)

Exercise 3.2 | Q 2.1 | Page 88

Find the polar co-ordinates of the point whose Cartesian co-ordinates are.

(sqrt(2), sqrt(2))

Exercise 3.2 | Q 2.2 | Page 88

Find the polar co-ordinates of the point whose Cartesian co-ordinates are.

(0, 1/2)

Exercise 3.2 | Q 2.3 | Page 88

Find the polar co-ordinates of the point whose Cartesian co-ordinates are.

(1, - √3)

Exercise 3.2 | Q 2.4 | Page 88

Find the polar co-ordinates of the point whose Cartesian co-ordinates are.

(3/2, (3√3)/2).

Exercise 3.2 | Q 3 | Page 88

In ΔABC, if ∠A = 45°, ∠B = 60° then find the ratio of its sides.

Exercise 3.2 | Q 4 | Page 88

In Δ ABC, prove that sin(("B - C")/2) = (("b - c")/2)cos  "A"/(2).

Exercise 3.2 | Q 5 | Page 88

With the usual notations prove that 2{asin^2  "C"/(2) + "c"sin^2  "A"/(2)} = a – b + c.

Exercise 3.2 | Q 6 | Page 88

In Δ ABC, prove that a3 sin(B – C) + b3sin(C – A) + c3sin(A – B) = 0

Exercise 3.2 | Q 7 | Page 88

In ΔABC, if cot A, cot B, cot C are in A.P. then show that a2, b2, c2 are also in A.P.

Exercise 3.2 | Q 8 | Page 88

In ΔABC, if a cos A = b cos B then prove that the triangle is either a right angled or an isosceles traingle.

Exercise 3.2 | Q 9 | Page 88

With usual notations prove that 2(bc cosA + ac cosB + ab cosC) = a2 + b2 + c.

Exercise 3.2 | Q 10.1 | Page 88

In ΔABC, if a = 18, b = 24, c = 30 then find the values of cosA

Exercise 3.2 | Q 10.2 | Page 88

In ΔABC, if a = 18, b = 24, c = 30 then find the values of sin A/2.

Exercise 3.2 | Q 10.3 | Page 88

In ΔABC, if a = 18, b = 24, c = 30 then find the values of cos A/2

Exercise 3.2 | Q 10.4 | Page 88

In ΔABC, if a = 18, b = 24, c = 30 then find the values of tan A/2

Exercise 3.2 | Q 10.5 | Page 88

In ΔABC, if a = 18, b = 24, c = 30 then find the values of A(ΔABC)

Exercise 3.2 | Q 10.6 | Page 88

In ΔABC, if a = 18, b = 24, c = 30 then find the values of sinA

Exercise 3.2 | Q 11 | Page 88

In ΔABC prove that (b + c - a) tan  "A"/(2) = (c + a - b)tan  "B"/(2) = (a + b - c)tan  "C"/(2).

Exercise 3.2 | Q 12 | Page 88

In ΔABC prove that sin  "A"/(2). sin  "B"/(2). sin  "C"/(2) = ["A(ΔABC)"]^2/"abcs"

Exercise 3.3 [Pages 102 - 103]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 3 Trigonometric Functions Exercise 3.3 [Pages 102 - 103]

Exercise 3.3 | Q 1.1 | Page 102

Find the principal value of the following: sin^-1 (1/2)

Exercise 3.3 | Q 1.2 | Page 102

Find the principal value of the following: cosec- 1(2)

Exercise 3.3 | Q 1.3 | Page 102

Find the principal value of the following: tan-1(– 1)

Exercise 3.3 | Q 1.4 | Page 102

Find the principal value of the following: tan- 1( - √3)

Exercise 3.3 | Q 1.5 | Page 102

Find the principal value of the following: sin-1 (1/sqrt(2))

Exercise 3.3 | Q 1.6 | Page 102

Find the principal value of the following: cos- 1(-1/2)

Exercise 3.3 | Q 2.1 | Page 102

Evaluate the following:

tan^-1(1) + cos^-1(1/2) + sin^-1(1/2)

Exercise 3.3 | Q 2.2 | Page 102

Evaluate the following:

cos^-1(1/2) + 2sin^-1(1/2)

Exercise 3.3 | Q 2.3 | Page 102

Evaluate the following:

tan^-1 sqrt(3) - sec^-1 (-2)

Exercise 3.3 | Q 2.4 | Page 103

Evaluate the following:

"cosec"^-1(-sqrt(2)) + cot^-1(sqrt(3))

Exercise 3.3 | Q 3.1 | Page 103

Prove the following:

sin^-1(1/sqrt(2)) -3sin^-1(sqrt(3)/2) = -(3pi)/(4)

Exercise 3.3 | Q 3.2 | Page 103

Prove the following:

sin^-1(-1/2) + cos^-1(-sqrt(3)/2) = cos^-1(-1/2)

Exercise 3.3 | Q 3.3 | Page 103

Prove the following:

sin^-1(3/5) + cos^-1(12/13) = sin^-1(56/65)

Exercise 3.3 | Q 3.4 | Page 103

Prove the following:

cos^-1(3/5) + cos^-1(4/5) = pi/(2)

Exercise 3.3 | Q 3.5 | Page 103

Prove the following:

tan^-1(1/2) + tan^-1(1/3) = pi/(4)

Exercise 3.3 | Q 3.6 | Page 103

Prove the following:

2tan^-1(1/3) = tan^-1(3/4)

Exercise 3.3 | Q 3.7 | Page 103

Prove the following:

tan^-1["cosθ + sinθ"/"cosθ - sinθ"] = pi/(4) + θ, if θ ∈ (- pi/4, pi/4)

Exercise 3.3 | Q 3.8 | Page 103

Prove the following:

tan^-1[sqrt((1 - cosθ)/(1 + cosθ))] = θ/(2), if θ ∈ (– π, π).

Miscellaneous exercise 3 [Pages 106 - 108]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 3 Trigonometric Functions Miscellaneous exercise 3 [Pages 106 - 108]

Miscellaneous exercise 3 | Q 1.01 | Page 106

Select the correct option from the given alternatives:

The principal solutions of equation sin θ = - 1/2 are

• (5pi)/6, pi/6

• (7pi)/6, (11pi)/6

• pi/6, (7pi)/6

• (7pi)/6, pi/3

Miscellaneous exercise 3 | Q 1.02 | Page 106

Select the correct option from the given alternatives:

The principal solutions of equation cot θ = sqrt3 are

• pi/6, (7pi)/6

• pi/6, (5pi)/6

• pi/6, (8pi)/6

• (7pi)/6, pi/3

Miscellaneous exercise 3 | Q 1.03 | Page 106

Select the correct option from the given alternatives:

The general solution of sec x = sqrt2 is

• 2"n"pi +- pi/4, "n" ∈ "Z"

• 2"n"pi +- pi/2, "n" ∈ "Z"

• "n"pi +- pi/2, "n" ∈ "Z"

• 2"n"pi +- pi/3, "n" ∈ "Z"

Miscellaneous exercise 3 | Q 1.04 | Page 106

Select the correct option from the given alternatives:

If cos pθ = cos qθ, p ≠ q, then,

• θ = (2"n"pi)/("p" +- "q")

• θ = 2nπ

• θ - 2nπ ± p

• θ = nπ ± q

Miscellaneous exercise 3 | Q 1.05 | Page 106

Select the correct option from the given alternatives:

If polar coordinates of a point are (2, pi/4), then its cartesian coordinates are

• (2, sqrt2)

• (sqrt2, 2)

• (2, 2)

• (sqrt2, sqrt2)

Miscellaneous exercise 3 | Q 1.06 | Page 106

Select the correct option from the given alternatives:

If sqrt3cos x - sin x = 1, then general value of x is

• 2"n"pi +- pi/3

• 2"n"pi +- pi/6

• 2"n"pi +- pi/3 - pi/6

• "n"pi + (- 1)^"n" pi/3

Miscellaneous exercise 3 | Q 1.07 | Page 107

Select the correct option from the given alternatives:

In Δ ABC if ∠A = 45°, ∠B = 60°, then the ratio of its sides are

• 2 : sqrt6 : sqrt3 + 1

• sqrt2 : 2 : sqrt3 + 1

• 2sqrt2 : sqrt2 : sqrt3

• 2 : 2sqrt2 : sqrt3 + 1

Miscellaneous exercise 3 | Q 1.08 | Page 107

Select the correct option from the given alternatives:

In ΔABC if c2 + a2 - b2 = ac, then ∠B = ____

• pi/4

• pi/3

• pi/2

• pi/6

Miscellaneous exercise 3 | Q 1.09 | Page 107

Select the correct option from the given alternatives:

In ΔABC, ac cos B - bc cos A = _______

• a2 - b2

• b2 - c2

• c2 - a2

• a2 - b2 - c2

Miscellaneous exercise 3 | Q 1.1 | Page 107

Select the correct option from the given alternatives:

If in a triangle, the angles are in A.P. and b: c = √3: √2, then A is equal to

• 30°

• 60°

• 75°

• 45°

Miscellaneous exercise 3 | Q 1.11 | Page 107

Select the correct option from the given alternatives:

"cos"^-1 ("cos" (7pi)/6) = _________.

• (7pi)/6

• (5pi)/6

• pi/6

• (3pi)/2

Miscellaneous exercise 3 | Q 1.12 | Page 107

Select the correct option from the given alternatives:

The value of cot (tan-12x + cot-12x) is

• 0

• 2x

• π + 2x

• π - 2x

Miscellaneous exercise 3 | Q 1.13 | Page 107

Select the correct option from the given alternatives:

The principal value of sin-1 (- sqrt3/2) is

• (- (2pi)/3)

• (4pi)/3

• (5pi)/3

• - pi/3

Miscellaneous exercise 3 | Q 1.14 | Page 107

Select the correct option from the given alternatives:

If "sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha, then α = ______

• 63/65

• 62/65

• 61/65

• 60/65

Miscellaneous exercise 3 | Q 1.15 | Page 107

Select the correct option from the given alternatives:

If tan-1(2x) + tan-1(3x) = pi/4, then x = _____

• - 1

• 1/6

• 2/6

• 3/2

Miscellaneous exercise 3 | Q 1.16 | Page 108

Select the correct option from the given alternatives:

2 "tan"^-1 (1/3) + "tan"^-1 (1/7) = _____

• "tan"^-1(4/5)

• pi/2

• 1

• pi/4

Miscellaneous exercise 3 | Q 1.17 | Page 108

Select the correct option from the given alternatives:

"tan"(2"tan"^-1 (1/5) - pi/4) = ______

• 17/7

• -17/7

• 7/17

• -7/17

Miscellaneous exercise 3 | Q 1.18 | Page 108

Select the correct option from the given alternatives:

The principal value branch of sec-1x is

• [-pi/2, pi/2] - {0}

• [0, pi] - {pi/2}

• (0, π)

• (- pi/2, pi/2)

Miscellaneous exercise 3 | Q 1.19 | Page 108

Select the correct option from the given alternatives:

"cos"["tan"^-1 1/3 + "tan"^-1 1/2] = ______

• 1/sqrt2

• sqrt3/2

• 1/2

• pi/4

Miscellaneous exercise 3 | Q 1.2 | Page 108

Select the correct option from the given alternatives:

If tan θ + tan 2θ + tan 3θ = tan θ.tan 2θ. tan 3θ, then the general value of the θ is

• ("n"pi)/6

• "n"pi +- pi/4

• ("n"pi)/2

Miscellaneous exercise 3 | Q 1.21 | Page 108

Select the correct option from the given alternatives:

In any ΔABC, if acos B = bcos A, then the triangle is

• equilateral triangle

• isosceles triangle

• scalene

• right-angled

Miscellaneous exercise 3 [Pages 108 - 111]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 3 Trigonometric Functions Miscellaneous exercise 3 [Pages 108 - 111]

Miscellaneous exercise 3 | Q 1.1 | Page 108

Find the principal solutions of the following equation:

sin 2θ = - 1/2

Miscellaneous exercise 3 | Q 1.2 | Page 108

Find the principal solutions of the following equation:

tan 3θ = - 1

Miscellaneous exercise 3 | Q 1.3 | Page 108

Find the principal solutions of the following equation:

cot θ = 0

Miscellaneous exercise 3 | Q 2.1 | Page 108

Find the principal solutions of the following equation:

sin 2θ = - 1/(√2).

Miscellaneous exercise 3 | Q 2.2 | Page 108

Find the principal solutions of the following equation:
tan 5θ = -1

Miscellaneous exercise 3 | Q 2.3 | Page 108

Find the principal solutions of the following equation:

cot 2θ = 0.

Miscellaneous exercise 3 | Q 3.1 | Page 109

State whether the following equation has a solution or not?

cos 2θ = 1/3

Miscellaneous exercise 3 | Q 3.2 | Page 109

State whether the following equation has a solution or not?

cos2θ = – 1.

Miscellaneous exercise 3 | Q 3.3 | Page 109

State whether the following equation has a solution or not?

2sinθ = 3

Miscellaneous exercise 3 | Q 3.4 | Page 109

State whether the following equation has a solution or not?

3 sin θ = 5.

Miscellaneous exercise 3 | Q 4.1 | Page 109

Find the general solutions of the following equation:

tan theta = - sqrt3

Miscellaneous exercise 3 | Q 4.2 | Page 109

Find the general solutions of the following equation:

tan^2 theta = 3

Miscellaneous exercise 3 | Q 4.3 | Page 109

Find the general solutions of the following equation:

sin θ - cos θ = 1

Miscellaneous exercise 3 | Q 4.4 | Page 109

Find the general solutions of the following equation:

sin2 θ - cos2 θ = 1

Miscellaneous exercise 3 | Q 5 | Page 109

In Δ ABC, prove that cos(("A" - "B")/2) = (("a" + "b")/"c")sin  "C"/2 .

Miscellaneous exercise 3 | Q 6 | Page 109

With the usual notations, prove that (sin("A" - "B"))/(sin ("A" + "B")) = ("a"^2 - "b"^2)/"c"^2

Miscellaneous exercise 3 | Q 7 | Page 109

In ΔABC, prove that ("a - b")^2 cos^2  "C"/2 + ("a + b")^2 sin^2  "C"/2 = "c"^2

Miscellaneous exercise 3 | Q 8 | Page 109

In Δ ABC, if cos A = sin B - cos C then show that it is a right-angled triangle.

Miscellaneous exercise 3 | Q 9 | Page 109

If (sin "A")/(sin "C") = (sin ("A - B"))/(sin ("B - C")), then show that a2, b2, c2 are in A.P.

Miscellaneous exercise 3 | Q 10 | Page 109

Solve the triangle in which a = (sqrt3 + 1), b = (sqrt3 - 1) and ∠C = 60°.

Miscellaneous exercise 3 | Q 11.1 | Page 109

In any Δ ABC, prove the following:

a sin A - b sin B = c sin (A - B)

Miscellaneous exercise 3 | Q 11.2 | Page 109

In any Δ ABC, prove the following:

("c" - "b cos A")/("b" - "c cos A") = ("cos B")/("cos C")

Miscellaneous exercise 3 | Q 11.3 | Page 109

In any Δ ABC, prove the following:

a2 sin (B - C) = (b2 - c2) sin A.

Miscellaneous exercise 3 | Q 11.4 | Page 109

In any Δ ABC, prove the following:

ac cos B - bc cos A = a2 - b2

Miscellaneous exercise 3 | Q 11.5 | Page 109

In any Δ ABC, prove the following:

"cos A"/"a" + "cos B"/"b" + "cos C"/"c" = ("a"^2 + "b"^2 + "c"^2)/(2"abc")

Miscellaneous exercise 3 | Q 11.6 | Page 109

In any Δ ABC, prove the following:

"cos 2A"/"a"^2 - "cos 2B"/"b"^2 = 1/"a"^2 - 1/"b"^2

Miscellaneous exercise 3 | Q 11.7 | Page 109

In any Δ ABC, prove the following:

("b" - "c")/"a" = (tan  "B"/2 - tan  "C"/2)/(tan  "B"/2 +tan  "C"/2)

Miscellaneous exercise 3 | Q 12 | Page 109

In Δ ABC, if a, b, c are in A.P., then show that cot "A"/2, cot  "B"/2, cot  "C"/2 are also in A.P.

Miscellaneous exercise 3 | Q 13 | Page 109

In Δ ABC, if ∠C = 90°, then prove that sin (A - B) = ("a"^2 - "b"^2)/("a"^2 + "b"^2)

Miscellaneous exercise 3 | Q 14 | Page 110

In Δ ABC, if "cos A"/"a" = "cos B"/"b", then show that it is an isosceles triangle.

Miscellaneous exercise 3 | Q 15 | Page 110

In Δ ABC, if sin2 A + sin2 B = sin2 C, then show that the triangle is a right-angled triangle.

Miscellaneous exercise 3 | Q 16 | Page 110

In Δ ABC, prove that a2 (cos2 B - cos2 C) + b2 (cos2 C - cos2 A) + c2 (cos2 A - cos2 B) = 0.

Miscellaneous exercise 3 | Q 17 | Page 110

With the usual notations, show that
(c2 - a2 + b2) tan A = (a2 - b2 + c2) tan B = (b2 - c2 + a2) tan C

Miscellaneous exercise 3 | Q 18 | Page 110

In Δ ABC, if a cos2 "C"/2 + "c cos"^2 "A"/2 = "3b"/2, then prove that a, b, c are in A.P.

Miscellaneous exercise 3 | Q 19 | Page 110

Show that 2 sin^-1 (3/5) = tan^-1(24/7)

Miscellaneous exercise 3 | Q 20 | Page 110

Show that

tan^-1(1/5) + tan^-1(1/7) + tan^-1(1/3) + tan^-1 (1/8) = pi/4.

Miscellaneous exercise 3 | Q 21 | Page 110

Prove that tan^-1 sqrt"x" = 1/2 cos^-1 ((1 - "x")/(1 + "x")), if x ∈ [0, 1]

Miscellaneous exercise 3 | Q 22 | Page 110

Show that (9pi)/8 - 9/4 sin^-1 (1/3) = 9/4 sin^-1 ((2sqrt2)/3).

Miscellaneous exercise 3 | Q 23 | Page 110

Show that tan^-1  ((sqrt(1 + "x") - sqrt(1 - "x"))/(sqrt(1 + "x") + sqrt(1 - "x"))) = pi/4 - 1/2 cos^-1 "x", for - 1/sqrt2 <= "x" <= 1

Miscellaneous exercise 3 | Q 24 | Page 110

If sin (sin^-1  1/5 + cos^-1 x) = 1, then find the value of x.

Miscellaneous exercise 3 | Q 25 | Page 110

If tan^-1 (("x" - 1)/("x" - 2)) + tan^-1 (("x" + 1)/("x" + 2)) = pi/4, find the value of x.

Miscellaneous exercise 3 | Q 26 | Page 110

If 2 tan-1(cos x) = tan-1(2 cosec x), then find the value of x.

Miscellaneous exercise 3 | Q 27 | Page 110

Solve: tan^-1 ("1 - x"/"1 + x") = 1/2 (tan^-1 "x"), for x > 0.

Miscellaneous exercise 3 | Q 28 | Page 110

If sin-1(1 - x) - 2 sin-1x =  pi/2, then find the value of x.

Miscellaneous exercise 3 | Q 29 | Page 110

If tan-12x + tan-13x = pi/4, then find the value of x.

Miscellaneous exercise 3 | Q 30 | Page 110

Show that tan^-1  1/2 - tan^-1  1/4 = tan^-1  2/9.

Miscellaneous exercise 3 | Q 31 | Page 110

Show that cot^-1  1/3 - tan^-1  1/3 = cot^-1  3/4.

Miscellaneous exercise 3 | Q 32 | Page 110

Show that tan^-1  1/2 = 1/3  tan^-1  11/2

Miscellaneous exercise 3 | Q 33 | Page 111

Show that cos^-1  sqrt3/2 + 2 sin^-1  sqrt3/2 = (5pi)/6.

Miscellaneous exercise 3 | Q 34 | Page 111

Show that 2 cot^-1  3/2 +  sec^-1  13/12 = pi/2

Miscellaneous exercise 3 | Q 35.1 | Page 111

Prove the following:

cos^-1 "x" = tan^-1 (sqrt(1 - "x"^2)/"x"), if x > 0

Miscellaneous exercise 3 | Q 35.2 | Page 111

Prove the following:

cos^-1 "x" = pi + tan^-1 (sqrt(1 - "x"^2)/"x"), if x < 0

Miscellaneous exercise 3 | Q 36 | Page 111

If | x | < 1, then prove that

2 tan^-1 "x" = tan^-1 ("2x"/(1 - "x"^2)) = sin^-1 ("2x"/(1 + "x"^2)) = cos^-1 ((1 - "x"^2)/(1 + "x"^2))

Miscellaneous exercise 3 | Q 37 | Page 111

If x, y, z are positive, then prove that

tan^-1 (("x - y")/(1 + "xy")) + tan^-1 (("y - z")/(1 + "yz")) + tan^-1 (("z - x")/(1 + "zx")) = 0

Miscellaneous exercise 3 | Q 38 | Page 111

If tan^-1 "x" + tan^-1 "y" + tan^-1 "z" = pi/2, then show that xy + yz + zx = 1

Miscellaneous exercise 3 | Q 39 | Page 111

If cos-1 x + cos-1y + cos-1z = 3π, then show that x2 + y2 + z2 + 2xyz = 1.

## Chapter 3: Trigonometric Functions

Exercise 3.1Exercise 3.2Exercise 3.3Miscellaneous exercise 3

## Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 3 - Trigonometric Functions

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