#### Online Mock Tests

#### Chapters

## Chapter 3: Trigonometric Functions

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

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 3 Trigonometric FunctionsExercise 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"`

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 3 Trigonometric FunctionsExercise 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).

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

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

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 3 (Trigonometric Functions) 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) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. Balbharati textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

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.

Using Balbharati 12th Board Exam solutions Trigonometric Functions exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Balbharati Solutions are important questions that can be asked in the final exam. Maximum students of Maharashtra State Board 12th Board Exam prefer Balbharati Textbook Solutions to score more in exam.

Get the free view of chapter 3 Trigonometric Functions 12th Board Exam extra questions for Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board and can use Shaalaa.com to keep it handy for your exam preparation