Find the exact value of the trigonometric expression given that sin(u) = − 3/5, where 3π/2 < u < 2π, and cos(v) = 15/17, where 0 < v < π/2. ? cos(u−v)

Question

Find the exact value of the trigonometric expression given that sin(u) = − 3/5, where 3π/2 < u < 2π, and cos(v) = 15/17, where 0 < v < π/2. ? cos(u−v)

Find the exact value of the trigonometric expression given that
sin(u) = − 3/5, where 3π/2 < u < 2π, and cos(v) = 15/17,
where 0 < v < π/2. ?

cos(u−v)

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Answer 1 · 2018-06-15T04:11:15

$cos (u - v) = \frac{36}{85}$ $cos (u - v) = 36/85$

Explanation:

Trig identity:
cos (u - v) = cos u.cos v + sin u.sin v (1)
$sin u = - \frac{3}{5}$ $sin u = - 3/5$ . Find cos u
${cos}^{2} u = 1 - {sin}^{2} u = 1 - \frac{9}{25} = \frac{16}{25}$ $cos^2 u = 1 - sin^2 u = 1 - 9/25 = 16/25$
$cos u = \frac{4}{5}$ $cos u = 4/5$ (because u lies in Quadrant 4)
$cos v = \frac{15}{17}$ $cos v = 15/17$ . Find sin v.
${sin}^{2} v = 1 - {cos}^{2} v = 1 - \frac{225}{289} = \frac{64}{289}$ $sin^2 v = 1 - cos^2 v = 1 - 225/289 = 64/289$
$sin v = \frac{8}{17}$ $sin v = 8/17$ (because v lies in Quadrant 1)
Replace these above values into identity (1)
$cos (u - v) = (\frac{15}{17}) (\frac{4}{5}) + (- \frac{3}{5}) (\frac{8}{17}) = \frac{36}{85}$ $cos (u - v) = (15/17)(4/5) + (-3/5)(8/17) = 36/85$

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Find the exact value of the trigonometric expression given that sin(u) = − 3/5, where 3π/2 < u < 2π, and cos(v) = 15/17, where 0 < v < π/2. ? cos(u−v)