Question #11152

find the values of √sinθ cosθ/secθ cscθ given that tan=3/4 and θ is an acute angle.

Expert's answer

Since θ is an acute angle, sinθ>0 and cosθ>0.

Then from tanθ=3/4 we get

that

sinθ = 3 / sqrt(3^2+4^2) = 3/5

cosθ = 4 / sqrt(3^2+4^2) =

4/5

Hence

secθ = 1/cosθ = 5/4

cscθ = 1/sinθ =

5/3

Therefore

sqrt(sinθ cosθ/secθ cscθ) =

= sqrt( (3/5 * 4/5)

/ (5/4 * 5/3) )

= sqrt( (3/5 * 4/5) * (3/5 * 4/5) )

= 3/5 *

4/5

= 12/25

Then from tanθ=3/4 we get

that

sinθ = 3 / sqrt(3^2+4^2) = 3/5

cosθ = 4 / sqrt(3^2+4^2) =

4/5

Hence

secθ = 1/cosθ = 5/4

cscθ = 1/sinθ =

5/3

Therefore

sqrt(sinθ cosθ/secθ cscθ) =

= sqrt( (3/5 * 4/5)

/ (5/4 * 5/3) )

= sqrt( (3/5 * 4/5) * (3/5 * 4/5) )

= 3/5 *

4/5

= 12/25

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