MathematicsA man on the top of a vertical tower observes a car moving at uniform speed coming directly towards it. If it takes 12 minutes for the angle of depression to change from 30° to 45°, how long will the car take to reach the observation tower from this point?                                                   (OR)Prove that tanΘ+sinΘtanΘ−sinθ=secΘ+1secΘ−1

A man on the top of a vertical tower observes a car moving at uniform speed coming directly towards it. If it takes 12 minutes for the angle of depression to change from 30° to 45°, how long will the car take to reach the observation tower from this point?

                                                   (OR)

Prove that tanΘ+sinΘtanΘsinθ=secΘ+1secΘ1

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    Solution:

     

     

     

     

     

     

     

     

     

     

     

    let the speed of the car be ‘𝑥’ 𝑚/𝑚𝑖𝑛 

    ⇒ 𝐵𝐶 = 𝑥 ✕12 

    In △ AOC

    tan30=AOOCAOOC=13OC=3AOOC=3OBAOOB=13AO=3OBOB+BC=3OBOB(31)=BCOBBC=131

    Let 𝑡 be the time taken for a car to reach the observation tower from the point Now, t=OBx

    t=BCx(31)t=1231×3+13+1t=12(3+1)31t=6(3+1)min

    Hence, the car takes 6 (3 +1)𝑚𝑖𝑛 to reach the observation tower from this point.

     

                                                 (OR)

     

     L.H.S =tanθ+sinθtanθsinθ=sinθcosθ+sinΘsinθcosθsinΘ=sinθ1cosθ+1sinθ1cosθ1=1cosθ+11cosθ1=secθ+1secθ1= R.H.S 

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