Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
TRIGONOMETRIC EQUATIONS PREVIOUS EAMCET BITS 1. If 3cos x ≠ 2,sin x , then the general solution of sin 2 x − cos 2x = 2 − sin 2x is x = [EAMCET 2009] 1) nπ + ( −1) n π ,n∈Z 2 2) π 3) ( 4n ± 1) , n ∈ Z 2 Ans: 3 nπ ,n∈Z 2 4) ( 2n − 1) π, n ∈ Z Sol. sin 2 x − (1 − 2sin 2 x ) = 2 − 2sin x cos x ⇒ 3sin 2 x + 2sin x cos x − 3 = 0 ⇒ 3sin 2 x + 2sin x cos x − 3 ( sin 2 x + cos 2 x ) = 0 ⇒ cos x ( 2sin x − 3cos x ) = 0 ⇒ cos x = 0 (∵ 2sin x ≠ 3cos x ) ⇒ x = ( 4n ± 1) π / 2, n ∈ Z 2. {x ∈ R : cos 2x + 2 cos 2 x − 2 = 0} = [EAMCET 2008] π π π ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ 1) ⎨2nπ + ; n ∈ Z⎬ 2) ⎨nπ ± ; n ∈ Z⎬ 3) ⎨nπ ± ; n ∈ Z⎬ 3 6 3 ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ Ans: 2 Sol. cos 2x + 2 cos 2 x − 2 = 0 ⇒ 2 cos 2 x − 1 + 2 cos 2 x − 2 = 0 3 π ⎛π⎞ ⇒ 4 cos 2 x = 3 ⇒ cos 2 x = = cos 2 ⎜ ⎟ ⇒ x = nπ ± 4 6 ⎝6⎠ ( ) 1 ⎞ 1 ⎛ 2 + 1 ⎜ cos x − ⎟ , cos x ≠ 2 ⇒ x ∈ 2⎠ ⎝ π π ⎧ ⎫ ⎧ ⎫ 2) ⎨2nπ ± : n ∈ Z ⎬ 1) ⎨2nπ ± : n ∈ Z ⎬ 3 6 ⎩ ⎭ ⎩ ⎭ π π ⎧ ⎫ ⎧ ⎫ 3) ⎨2nπ ± : n ∈ Z⎬ 4) ⎨2nπ ± : n ∈ Z⎬ 2 4 ⎩ ⎭ ⎩ ⎭ Ans: 4 Sol. 2 2 cos 2 x − 2 + 2 cos x + 1 = 0 3. cos 2x = ( ) 2 2 cos x − 2 cos x − 2 cos x + 1 = 0 2 ( 2 cos x − 1) ( cos x ≠ 1 ; 2 ) 2 cos x − 1 = 0 cos x = 1 2 1 π ⎧ ⎫ 4) ⎨2nπ − ; n ∈ Z⎬ 3 ⎩ ⎭ [EAMCET 2005] Trigonometric Equations 4. π ⎧ ⎫ ∴ x ∈ ⎨ 2nπ ± / n ∈ Z ⎬ 4 ⎩ ⎭ The solution set of ( 5 + 4 cos θ )( 2 cos θ + 1) = 0 in the interval [0, 2π] is ⎧ π 2π ⎫ ⎧ 2 π 5π ⎫ 2) ⎨ , ⎬ 1) ⎨ , ⎬ ⎩3 3 ⎭ ⎩3 3⎭ Ans: 3 Sol. ( 5 + 4 cos θ )( 2 cos θ + 1) = 0 2π 4π −1 , ⇒θ= 2 3 3 5. The equation 3 sin x + cos x = 4 has 1) only one solution 3) infinitely many solutions Ans: 4 Sol. The max. value of 3 sinx + cosx is ‘2’. ∴ 3 sinx + cosx never equal to ‘4’. ⎧ 2π 4 π ⎫ 3) ⎨ , ⎬ ⎩3 3⎭ [EAMCET 2003] ⎧ 2 π 5π ⎫ 4) ⎨ , ⎬ ⎩3 3⎭ cos θ = 6. [EAMCET 2001] 2) two solutions 4) no solutions If tan θ + sec θ = 3 , then the principal value of θ + 1) π/4 Ans: 2 Sol. tan θ + sec θ = 3 2) π/3 3) 2π/3 3 cos θ − sin θ = 1 π⎞ 1 π⎞ π ⎛ ⎛ ∴⎜θ+ ⎟ = cos ⎜ θ + ⎟ = 6⎠ 2 6⎠ 3 ⎝ ⎝ 2 π is 6 [EAMCET 2000] 4) 3π/4