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From a point on a hillside of constant i...

From a point on a hillside of constant inclination , the angle of elevation of the top a flagstaff on its summit is observed to be `alpha` and a meters nears the top of the hill, it is `beta` .If h is the height of the flagstaff ,find the inclination of the hill to the horizon .

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Let OP be the flagstaff of height h and `theta` inclination of the hill.

`Then angle ORP = alpha,angle OQP=beta and RQ =a.`
Also,`angle RPQ =beta-alpha .`
Now ,in triangle PQR ,using sine rule ,we get
`pr/(sin(180^@-beta))=(RQ)/sin(beta-alpha)`
`rArr PR = (asinbeta) /sin(beta-alpha) ` ...(1)
Also, `anglePOR= pi/2+ theta `
Now , in triangle PQR ,using sine rule , we get
`(PO)/sinalpha=(PR)/sin(pi/2+theta)`
`rArr PR =(hcostheta)/(sinalpha)=(asinbeta)/sin(beta-alpha)` (using(1))
`rArr theta = cos^-1[(asinalpha sin beta )/(hsin(beta-alpha))]`
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