#### Chapters

## Chapter 3: Trigonometric Functions

#### 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]

Find the principal solution of the following equation:

cosθ = `(1)/(2)`

Find the principal solution of the following equation:

Secθ = `(2)/sqrt(3)`

Find the principal solution of the following equation :

cotθ = √3

Find the principal solution of the following equation:

cotθ = 0

Find the principal solution of the following equation:

sin θ = `-1/2`

Find the principal solution of the following equation:

tan θ = – 1

Find the principal solution of the following equation:

`sqrt(3)`cosecθ+ 2 = 0

Find the general solution of the following equation:

sinθ = `1/2`.

Find the general solution of the following equation :

cosθ = `sqrt(3)/(2)`

Find the general solution of the following equation:

tan θ = `(1)/(sqrt(3))`

Find the general solution of the following equation:

cot θ = 0.

Find the general solution of the following equation:

sec θ = `sqrt(2)`.

Find the general solution of the following equation:

cosec θ = - √2.

Find the general solution of the following equation:

tan θ = - 1

Find the general solution of the following equation:

sin 2θ = `1/2`

Find the general solution of the following equation:

tan `(2θ)/(3)` = √3.

Find the general solution of the following equation:

cot 4θ = – 1

Find the general solution of the following equation:

4cos^{2}θ = 3.

Find the general solution of the following equation:

4sin^{2}θ = 1.

Find the general solution of the following equation:

cos 4θ = cos 2θ

Find the general solution of the following equation:

sin θ = tan θ.

Find the general solution of the following equation:

tan^{3}θ = 3 tanθ.

Find the general solution of the following equation:

cos θ + sin θ = 1.

State whether the following equation have solution or not?

cos 2θ = – 1

**State whether the following equation has a solution or not?**

cos^{2}θ = – 1.

**State whether the following equation has a solution or not?**

2sinθ = 3

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]

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

`(sqrt(2), pi/4)`

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

`(4, pi/2)`

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

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

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

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

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

`(sqrt(2), sqrt(2))`

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

`(0, 1/2)`

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

(1, - √3)

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

`(3/2, (3√3)/2)`.

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

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

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

In Δ ABC, prove that a^{3} sin(B – C) + b^{3}sin(C – A) + c^{3}sin(A – B) = 0

In ΔABC, if cot A, cot B, cot C are in A.P. then show that a^{2}, b^{2}, c^{2} are also in A.P.

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

With usual notations prove that 2(bc cosA + ac cosB + ab cosC) = a^{2} + b^{2} + c^{2 }.

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

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

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

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

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

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

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

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]

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

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

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

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

Find the principal value of the following: sin^{-1} `(1/sqrt(2))`

Find the principal value of the following: cos^{- 1}`(-1/2)`

Evaluate the following:

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

Evaluate the following:

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

Evaluate the following:

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

Evaluate the following:

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

Prove the following:

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

Prove the following:

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

Prove the following:

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

Prove the following:

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

Prove the following:

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

Prove the following:

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

Prove the following:

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

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]

**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`

**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`

**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"`

**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

**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)`

**Select the correct option from the given alternatives:**

If `sqrt3`cos 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`

**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`

**Select the correct option from the given alternatives:**

In ΔABC if c^{2} + a^{2} - b^{2} = ac, then ∠B = ____

`pi/4`

`pi/3`

`pi/2`

`pi/6`

**Select the correct option from the given alternatives:**

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

a

^{2}- b^{2}b

^{2}- c^{2}c

^{2}- a^{2}a

^{2}- b^{2}- c^{2}

**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°

**Select the correct option from the given alternatives:**

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

`(7pi)/6`

`(5pi)/6`

`pi/6`

`(3pi)/2`

**Select the correct option from the given alternatives:**

The value of cot (tan^{-1}2x + cot^{-1}2x) is

0

2x

π + 2x

π - 2x

**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`

**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`

**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`

**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`

**Select the correct option from the given alternatives:**

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

`17/7`

`-17/7`

`7/17`

`-7/17`

**Select the correct option from the given alternatives:**

The principal value branch of sec^{-1}x is

`[-pi/2, pi/2] - {0}`

`[0, pi] - {pi/2}`

(0, π)

`(- pi/2, pi/2)`

**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`

**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

nπ

`("n"pi)/6`

`"n"pi +- pi/4`

`("n"pi)/2`

**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]

**Find the principal solutions of the following equation:**

sin 2θ = - `1/2`

**Find the principal solutions of the following equation:**

tan 3θ = - 1

**Find the principal solutions of the following equation:**

cot θ = 0

**Find the principal solutions of the following equation:**

sin 2θ = `- 1/(√2)`.

**Find the principal solutions of the following equation:**

tan 5θ = -1

**Find the principal solutions of the following equation:**

cot 2θ = 0.

**State whether the following equation has a solution or not?**

cos 2θ = `1/3`

**State whether the following equation has a solution or not?**

cos^{2}θ = – 1.

**State whether the following equation has a solution or not?**

2sinθ = 3

**State whether the following equation has a solution or not?**

3 sin θ = 5.

**Find the general solutions of the following equation:**

`tan theta = - sqrt3`

**Find the general solutions of the following equation:**

`tan^2 theta = 3`

**Find the general solutions of the following equation:**

sin θ - cos θ = 1

**Find the general solutions of the following equation:**

sin^{2} θ - cos^{2} θ = 1

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

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

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

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

If `(sin "A")/(sin "C") = (sin ("A - B"))/(sin ("B - C"))`, then show that a^{2}, b^{2}, c^{2} are in A.P.

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

**In any Δ ABC, prove the following:**

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

**In any Δ ABC, prove the following:**

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

**In any Δ ABC, prove the following:**

a^{2} sin (B - C) = (b^{2} - c^{2}) sin A.

**In any Δ ABC, prove the following:**

ac cos B - bc cos A = a^{2} - b^{2}

**In any Δ ABC, prove the following:**

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

**In any Δ ABC, prove the following:**

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

**In any Δ ABC, prove the following:**

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

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.

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

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

In Δ ABC, if sin^{2} A + sin^{2} B = sin^{2} C, then show that the triangle is a right-angled triangle.

In Δ ABC, prove that a^{2} (cos^{2} B - cos^{2} C) + b^{2} (cos^{2} C - cos^{2} A) + c^{2} (cos^{2} A - cos^{2} B) = 0.

With the usual notations, show that

(c^{2} - a^{2} + b^{2}) tan A = (a^{2} - b^{2} + c^{2}) tan B = (b^{2} - c^{2} + a^{2}) tan C

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

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

Show that

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

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

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

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`

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

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

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

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

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

If tan^{-1}2x + tan^{-1}3x = `pi/4`, then find the value of x.

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

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

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

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

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

**Prove the following:**

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

**Prove the following:**

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

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))`

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`

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

If cos^{-1} x + cos^{-1}y + cos^{-1}z = 3π, then show that x^{2} + y^{2} + z^{2} + 2xyz = 1.

## Chapter 3: Trigonometric Functions

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

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Concepts covered in Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 3 Trigonometric Functions are Trigonometric Equations and Their Solutions, Solutions of Triangle, Inverse Trigonometric Functions.

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