3 an 36+ sin17524. Sin

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Solution

We have to prove that sin −1 8 17 + sin −1 3 5 = tan −1 77 36 .

Consider sin −1 8 17 =x, then,

sinx= 8 17 cosx= 1− ( 8 17 ) 2 = 225 289 = 15 17

Another trigonometric function is,

tanx= sinx cosx = 8 15 x= tan −1 8 15 sin −1 8 17 = tan −1 8 15

Consider sin −1 3 5 =y, then,

siny= 3 5 cosy= 1− ( 3 5 ) 2 = 16 25 = 4 5

Another trigonometric function is,

tany= siny cosy = 3 4 y= tan −1 3 4 sin −1 3 5 = tan −1 3 4

Substitute sin −1 8 17 = tan −1 8 15 and sin −1 3 5 = tan −1 3 4 to the left hand side of the given equation,

sin −1 8 17 + sin −1 3 5 = tan −1 8 15 + tan −1 3 4 = tan −1 ( 8 15 + 3 4 1− 8 15 × 3 4 ) = tan −1 32+45 60−24 = tan −1 77 36

Hence, it is proved that sin −1 8 17 + sin −1 3 5 = tan −1 77 36 .

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sin \frac{17 π}{36} cos \frac{11 π}{36} cos \frac{13 π}{36} sin \frac{11 π}{36} sin \frac{13 π}{36} cos \frac{17 π}{36}

sin \frac{17 π}{36} cos \frac{11 π}{36} cos \frac{13 π}{36} sin \frac{11 π}{36} sin \frac{13 π}{36} cos \frac{17 π}{36}

θ

(θ + 36^{\circ}) = cos θ

θ + 36^{\circ}

Prove, $s i n^{- 1} \frac{8}{17} + s i n^{- 1} \frac{3}{5} = t a n^{- 1} \frac{77}{36}$

{sin}^{- 1} \frac{3}{5} + {sin}^{- 1} \frac{8}{17} = {cos}^{- 1} \frac{36}{85}

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