General Solution of sin theta = sin alpha

Q.

If ${\cos }^{-1}x>{\sin }^{-1}x$, then

1. $\frac{1}{\sqrt{2}}<x⩽1$

2. $0<x<\frac{1}{\sqrt{2}}$

3. $-1\le x<\frac{1}{\sqrt{2}}$

4. $x>0$

Q. Solve the following trigonometric equations when 0⁰ <=ᶿ <= 90⁰
​​

1. Cos ²ᶿ - 3 cos ᶿ + 2 = 2 sin ² ᶿ 0⁰ <ᶿ≤ 90⁰

Q. The general solution of the equation $7{\cos }^2\mathrm{\theta }+3{\sin }^2\mathrm{\theta }=4$ is
(a) $\mathrm{\theta }=2n\pi \pm \frac{\pi }{6},n\in Z$

(b) $\mathrm{\theta }=2n\pi \pm \frac{2\pi }{3},n\in Z$

(c) $\mathrm{\theta }=n\pi \pm \frac{\pi }{3},n\in Z$

(d) none of these

Q. If $e^{\sin x}-e^{-\sin x}-4=0$, then x =
(a) 0
(b) ${\sin }^{-1}\left\{{\log }_e\left(2-\sqrt{5}\right)\right\}$
(c) 1
(d) none of these

Q. The general solution of the trigonometrical equation $\sin x+\cos x=1$, for n = 0 , $\pm$1, $\pm$2, .......is given by

1. x = $2n\pi +\frac{1}{2}\pi$
2. None of these
3. x= $n\pi +(-1{)}^n\frac{\pi }{4}-\frac{\pi }{4}$
4. x= $2n\pi$

Q. Solve the following equations:
(i) $\cot x+\tan x=2$ [NCERT EXEMPLAR]
(ii) $2{\sin }^{2}x=3\cos x,0\le x\le 2\mathrm{\pi }$ [NCERT EXEMPLAR]
(iii) $\sec x\cos 5x+1=0,0<x<\frac{\mathrm{\pi }}{2}$ [NCERT EXEMPLAR]
(iv) $5{\cos }^{2}x+7{\sin }^{2}x-6=0$ [NCERT EXEMPLAR]
(v) $\sin x-3\sin 2x+\sin 3x=\cos x-3\cos 2x+\cos 3x$ [NCERT EXEMPLAR]
(vi) 4sinx cosx + 2 sin x + 2 cosx + 1 = 0
(vii) cosx + sin x = cos 2x + sin 2x
(viii) sin x tan x – 1 = tan x – sin x
(ix) 3tanx + cot x = 5 cosec x

Q. Write the number of values of θ in [0, 2π] that satisfy the equation $\sin \mathrm{\theta }-\cos \mathrm{\theta }=\frac{1}{4}$.

Q. Solve the following equations:
(i) $\cos \mathrm{\theta }+\cos 2\mathrm{\theta }+\cos 3\mathrm{\theta }=0$
(ii) $\cos \mathrm{\theta }+\cos 3\mathrm{\theta }-\cos 2\mathrm{\theta }=0$
(iii) $\sin \mathrm{\theta }+\sin 5\mathrm{\theta }=\sin 3\mathrm{\theta }$
(iv) $\cos \mathrm{\theta }\cos 2\mathrm{\theta }\cos 3\mathrm{\theta }=\frac{1}{4}$
(v) $\cos \mathrm{\theta }+\sin \mathrm{\theta }=\cos 2\mathrm{\theta }+\sin 2\mathrm{\theta }$
(vi) $\sin \mathrm{\theta }+\sin 2\mathrm{\theta }+\sin 3=0$
(vii) $\sin \mathrm{\theta }+\sin 2\mathrm{\theta }+\sin 3\mathrm{\theta }+\sin 4\mathrm{\theta }=0$
(viii) $\sin 3\mathrm{\theta }-\sin \mathrm{\theta }=4{\cos }^2\mathrm{\theta }-2$
(ix) $\sin 2\mathrm{\theta }-\sin 4\mathrm{\theta }+\sin 6\mathrm{\theta }=0$

Q.

Find the exact value of ${\sin }^{-1}(-0.5)$.

Q. The general solution for $\sin x=\frac{\sqrt{3}}{2}$ is given as

1. $2n\pi +\frac{\pi }{3}$
2. $2n\pi -\frac{\pi }{3}$
3. $n\pi +(-1{)}^n\frac{\pi }{3}$
4. $2n\pi +(-1{)}^n\frac{\pi }{3}$

Q. Solve the following equations:
(i) ${\sin }^2\mathrm{\theta }-\cos \mathrm{\theta }=\frac{1}{4}$
(ii) $2{\cos }^2\mathrm{\theta }-5\cos \mathrm{\theta }+2=0$
(iii) $2{\sin }^{2}x+\sqrt{3}\cos x+1=0$
(iv) $4{\sin }^2\mathrm{\theta }-8\cos \mathrm{\theta }+1=0$
(v) ${\tan }^{2}x+\left(1-\sqrt{3}\right)\tan x-\sqrt{3}=0$
(vi) $3{\cos }^2\mathrm{\theta }-2\sqrt{3}\sin \mathrm{\theta }\cos \mathrm{\theta }-3{\sin }^2\mathrm{\theta }=0$
(vii) $\cos 4\mathrm{\theta }=\cos 2\mathrm{\theta }$

Q. If $\sin \theta ,1,\cos 2\theta$ are in G.P., then

1. $\theta =2n\pi +(-1{)}^n\frac{\pi }{2},n\in Z$
2. $\theta =n\pi -\frac{\pi }{2},n\in Z$
3. $\theta =2n\pi -\frac{\pi }{2},n\in Z$
4. $\theta =n\pi +(-1{)}^n\frac{\pi }{2},n\in Z$

Q. Find the general solution of the equation :
​tan x + tan (​​pi/3 + x) + tan (2pi/3 + x) = 3

Q. The equation $3\cos x+4\sin x=6$ has .... solution.
(a) finite
(b) infinite
(c) one
(d) no

Q. The number of solutions of the equation $\sin \left(\frac{\pi x}{2\sqrt{3}}\right)=x^2-2\sqrt{3}x+4$

1. forms an empty set
2. is only one
3. is only two
4. is greater than $2$

Q. The number of values of θ in [0, 2π] that satisfy the equation ${\sin }^2\mathrm{\theta }-\cos \mathrm{\theta }=\frac{1}{4}$
(a) 1
(b) 2
(c) 3
(d) 4

Q. For $0\le x\le 2\pi$, then $2^{{\text{cosec}}^{2}x}\sqrt{\frac{1}{2}{y}^2-y+1}\le \sqrt{2}$

1. is satisfied by exactly one value of $y$
2. is satisfied by exactly two values of $x$
3. is satisfied by $x$ for which $\cos x=\frac{1}{2}$
4. is satisfied by $x$ for which $\sin x=0$

Q. The solution of the equation ${\cos }^2\mathrm{\theta }+\sin \mathrm{\theta }+1=0$ lies in the interval
(a) $\left(-\pi /4,\pi /4\right)$
(b) $\left(\pi /4,3\pi /4\right)$
(c) $\left(3\pi /4,5\pi /4\right)$
(d) $\left(5\pi /4,7\pi /4\right)$

Q. If $\sin 2\theta \cos 2\theta =-\frac{1}{4},$ then the values of $\theta$ which satisfy the equation is

1. $\theta =\frac{n\pi }{8}+(-1{)}^n(-\frac{\pi }{48}),n\in Z$
2. $\theta =\frac{n\pi }{4}+(-1{)}^n(-\frac{\pi }{24}),n\in Z$
3. $\theta =\frac{n\pi }{2}+(-1{)}^n(-\frac{\pi }{24}),n\in Z$
4. $\theta =\frac{n\pi }{4}+(-1{)}^n(-\frac{\pi }{12}),n\in Z$

Q. The general solution of the equation $2{\cos }^2\theta +3\sin \theta =0$ is

1. $n\pi +(-1{)}^{n+1}\frac{\pi }{6},n\in Z$
2. $n\pi +(-1{)}^n\frac{5\pi }{6},n\in Z$
3. $n\pi +(-1{)}^n\frac{\pi }{6},n\in Z$
4. None of these