The number of values of $$x$$ in the interval $$\left[ {0,3\pi } \right]\,$$ satisfying the equation $$2{\sin ^2}x + 5\sin x - 3 = 0$$ is
A
4
B
6
C
1
D
2
Explanation
$$2{\sin ^2}x + 5\sin x - 3 = 0$$
$$ \Rightarrow \left( {\sin x + 3} \right)\left( {2\sin x - 1} \right) = 0$$
$$\sin x = {1 \over 2}$$ and $$\,\,\sin x \ne - 3$$
Given that $$x \in \left[ {0,3\pi } \right]$$
So possible values of x are $$30^\circ $$, $$150^\circ $$, $$390^\circ $$, $$510^\circ $$. That means x have 4 values.
4
AIEEE 2004
MCQ (Single Correct Answer)
A line makes the same angle $$\theta $$, with each of the $$x$$ and $$z$$ axis.
If the angle $$\beta \,$$, which it makes with y-axis, is such that $$\,{\sin ^2}\beta = 3{\sin ^2}\theta ,$$ then $${\cos ^2}\theta $$ equals
A
$${2 \over 5}$$
B
$${1 \over 5}$$
C
$${3 \over 5}$$
D
$${2 \over 3}$$
Explanation
Concept : If a line makes the angle $$\alpha ,\beta ,\gamma $$ with x, y, z axis respectively then
$$${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1$$$
In this question given that the line makes angle θ with x and z-axis and β with y−axis.