Let `P M` be the perpendicular from the point `P(1,2,3)` to the `x-y` plane. If ` vec O P` makes an angle `theta` with the positive direction of the `z-` axis and` vec O M` makes an angle `varphi` with the positive direction of `x-` axis, `w h e r eO` is the origin and `thetaa n dvarphi` are acute angels, then a. `costhetacosvarphi=1//sqrt(14)` b. `sinthetasinvarphi=2//sqrt(14)` c. `""tanvarphi=2` d. `tantheta=sqrt(5)//3`

Options

A. `costhetacosphi=1//sqrt14`
B. `sinthetasinphi=2//sqrt14`
C. `tanphi=2`
D. `tantheta=sqrt5//3`

Answer

b, c, d

Explanation

If `P` be `(x, y, z),` then from the figure,
`x=r sin theta cosphi, y = rsintheta sin phi and z= rcostheta`
`1=rsintheta cosphi, 2= r sintheta sinphi and 3 = rcostheta`

`rArr" "1^(2)+ 2^(2)+3^(2)= r^(2) or r= pm sqrt(14)`
`therefore" "sinthetacosphi = (1)/(sqrt(14)), sin theta sin phi`
`" "=(2)/(sqrt(14)) and cos theta = (3)/(sqrt(14)) `
(neglecting negative sign as `theta and phi` are acute)
`" "(sinthetasinphi)/(sinthetacosphi)=(2)/(1) or tanphi =2`
Also, `tan theta= sqrt(5)//3`.