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

Find the principal solution of the following equation: 

cosθ = `(1)/(2)`

Find the principal solution of the following equation: 

Sec θ = `(2)/sqrt(3)`

Find the principal solution of the following equation :

cot θ = `sqrt(3)`

Find the principal solution of the following equation:

cot θ = 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) = sqrt3`.

Find the general solution of the following equation:

cot 4θ = – 1

Find the general solution of the following equation:

4cos²θ  = 3.

Find the general solution of the following equation:

4sin²θ = 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: 

tan³θ = 3 tanθ.

Find the general solution of the following equation:

cosθ + sinθ = 1.

State whether the following equation have solution or not?

cos 2θ = – 1

State whether the following equation has a solution or not?

cos²θ = – 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

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

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

Find the Cartesian coordinates of the point whose polar coordinates 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, - sqrt(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 ΔABC, prove that `sin(("B" − "C")/2) = (("b" − "c")/"a")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 a³ sin(B – C) + b³sin(C – A) + c³sin(A – B) = 0

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

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"`

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) = -(3π)/(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).

Select the correct option from the given alternatives:

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

Select the correct option from the given alternatives:

The principal solutions of equation cot θ = `sqrt3` are ______.

Select the correct option from the given alternatives:

The general solution of sec x = `sqrt(2)` is ______.

Select the correct option from the given alternatives:

If cos pθ = cos qθ, p ≠ q, then ______.

Select the correct option from the given alternatives:

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

If `sqrt3`cos x - sin x = 1, then general value of x is ______.

Select the correct option from the given alternatives:

In Δ ABC if ∠A = 45°, ∠B = 30°, then the ratio of its sides are

Select the correct option from the given alternatives:

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

Select the correct option from the given alternatives:

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

If in a triangle, the angles are in A.P. and b: c = `sqrt3: sqrt2`, then A is equal to

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

The principal value of sin^–1 `(- sqrt3/2)` is ______.

Select the correct option from the given alternatives:

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

Select the correct option from the given alternatives:

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

Select the correct option from the given alternatives:

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

Select the correct option from the given alternatives:

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

Select the correct option from the given alternatives:

The principal value branch of sec^-1x is

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

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

Select the correct option from the given alternatives:

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

Select the correct option from the given alternatives:

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

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/(sqrt2)`

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?

cos²θ = – 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:

sin² θ - cos² 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 a², b², c² 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 ΔABC, prove the following:

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

In any Δ ABC, prove the following:

a² sin (B - C) = (b² - c²) sin A.

In any Δ ABC, prove the following:

ac cos B - bc cos A = a² - b²

In ΔABC, prove the following:

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

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

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

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

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

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.

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)`.

Prove 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`

[Hint: put x =  cos 2θ]

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^-1x =  `pi/2`, then find the value of x.

If tan^-12x + tan^-13x = `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 = π/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^-1y + cos^-1z = 3π, then show that x² + y² + z² + 2xyz = 1.