SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC chapter 1.3 - Trigonometric Functions [Latest edition]

The principal solutions of `sqrt(3)` sec x − 2 = 0 are ______

In ∆ABC, if cos A = `(sinB)/(2sinC)`, then ∆ABC is ______.

sin⁻¹x − cos⁻¹x = `pi/6`, then x = ______

The principal value of sin⁻¹`(1/2)` is ______

The principal value of cos⁻¹`(-1/2)` is ______

In ∆ABC, if ∠A = 30°, ∠B = 60°, then the ratio of sides is ______.

In ∆ABC, if b² + c² − a² = bc, then ∠A = ______.

If polar co-ordinates of a point are `(3/4, (3pi)/4)`, then its Cartesian co-ordinate are ______

`tan^-1(tan  (7pi)/6)` = ______

If `sin(sin^-1(1/5) + cos^-1(x))` = 1, then x = ______

Evaluate cot(tan⁻¹(2x) + cot⁻¹(2x))

In ∆ABC, prove that ac cos B − bc cos A = a² − b² 

In ∆ABC, if sin²A + sin²B = sin²C, then show that a² + b² = c² 

Find the polar co-ordinates of point whose Cartesian co-ordinates are `(1, sqrt(3))`

Prove that `2 tan^-1 (3/4) = tan^-1(24/7)`

Evaluate:

`sin[cos^-1 (3/5)]`

In ΔABC, a = 3, b = 4 and sin A = `3/4`, find ∠B

Find the principal solutions of cosec x = 2

Find the principal solutions of sin x − 1 = 0

Find the Cartesian co-ordinates of point whose polar co-ordinates are `(4, pi/3)`

With usual notations, prove that `(cos "A")/"a" + (cos "B")/"b" + (cos "C")/"c" = ("a"^2 + "b"^2 + "c"^2)/(2"abc")`

Find the principal solutions of cos 2𝑥 = 1

In ∆ABC, prove that `("b" - "c")^2 cos^2 ("A"/2) + ("b" + "c")^2 sin^2 ("A"/2)` = a² 

Find the principal solutions of sin x = `-1/2`

Find the value of `cos^-1 (1/2) + tan^-1 (1/sqrt(3))`

In ∆ABC, if a = 13, b = 14, c = 15, then find the value of cos B

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

Find the principal solutions of tan x = `-sqrt(3)`

Evaluate `cos[pi/6 + cos^-1 (- sqrt(3)/2)]`

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

In ∆ABC, prove that `(cos 2"A")/"a"^2 - (cos 2"c")/"c"^2 = 1/"a"^2 - 1/"c"^2`

If tan⁻¹x + tan⁻¹y + tan⁻¹z = π, then show that `1/(xy) + 1/(yz) + 1/(zx)` = 1

Prove that sin `[tan^-1 ((1 - x^2)/(2x)) + cos^-1 ((1 - x^2)/(1 + x^2))]` = 1

In ∆ABC, if `(2cos "A")/"a" + (cos "B")/"b" + (2cos"C")/"c" = "a"/"bc" + "b"/"ca"`, then show that the triangle is a right angled

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

If the angles A, B, C of ΔABC are in A.P. and its sides a, b, c are in G.P., then show that a², b², c² are in A.P.

Prove that cot⁻¹(7) + 2 cot⁻¹(3) = `pi/4`

In ∆ABC, prove that `(cos^2"A" - cos^2"B")/("a" + "b") + (cos^2"B" - cos^2"C")/("b" + "c") + (cos^2"C" - cos^2"A")/("c" + "a")` = 0

Show that `sin^-1(3/5)  + sin^-1(8/17) = cos^-1(36/85)`

In ΔABC, prove that `("a"^2sin("B" - "C"))/(sin"A") + ("b"^2sin("C" - "A"))/(sin"B") + ("c"^2sin("A" - "B"))/(sin"C")` = 0

In ΔABC, prove that `("b"^2 - "c"^2)/"a" cos"A" + ("c"^2 - "a"^2)/"b" cos"B" + ("a"^2 - "b"^2)/"c" cos "C"` = 0

Prove that `2 tan^-1 (1/8) + tan^-1 (1/7) + 2tan^-1 (1/5) = pi/4`

In ∆ABC, if ∠A = `pi/2`, then prove that sin(B − C) = `("b"^2 - "c"^2)/("b"^2 + "c"^2)`

If cos^–1x + cos^–1y – cos^–1z = 0, then show that x² + y² + z² – 2xyz = 1