Trigonometric functions:class 12th: DPP

Trigonometric Functions

Q 1 Answer the following questions                                 (5M)

a) The solutions of a trigonometric equation which are not generalized by using its periodicity are known as _____.

i) General Solution  ii) Principle Solution iii) Normal Solution iv) Ordinary Solution

b) Which of the following is not a trigonometric equation.

i) sinx = ⎷3/2      ii) secӨ = cosec x   iii) ⎷3/x = 17  iv)  cosx + Sin 3x + tan 5x = 0

d) If cos Ө = 0 then Ө = _______

i) nπ ii) (2n+1)π/2   iii) 2nπ  iv) (2n+1)π

e) For Cos x = ½ then the general solution of x is

i)0  ii) π/3,5π/3  iii) π/3,5π/3,13π/3….. iv) (n+1) π/3

Q2 Prove the following equations (any three)         (6M)

a) If tan ²x =  tan ²β then x = nπ 士 β

b) Find the Principle solution of cos3x = cos2x

c) Find the general Solution of tan 3x = -1

d) Find the general solution of ⎷3cosec x = 2

Q3 Find the general Solution of equation            (9M)

a) cos5x = sin3x

b) cosx - sinx = 1

c) 2tanx - cotx + 1 = 0
Q4)  Show that                                                       (6M)

a) 2tan^-1x = cos^-1(1-x²/1+x²) For x>0

b) 2tan^-1x =  tan^-1(2x/1-x²) For -1

Q5) Prove the following properties                      (6M)

a) sin^-1(x) + cos^-1(x) = π/2  for x∈[-1,1]

b) tan^-1(x) + tan^-1(y) = tan^-1((x+y)/(1-xy)),if x,y>0 and xy≠1

Q6) Prove the property                                         (4M)

a) cosec^-1(-x) = - cosec^-1(x)

b) cot^-1(-x) = π - cot^-1(x)

Q7) Show that                                                          (4M)

tan^-1x + tan^-1(2x/1+x²) = tan^-1(3x - x³/1-3x²).     for |x|