The evaluation of sin(x+y) will be equal to sin(x+y) =(3√5 + 8)/15
What are trigonometric identities?
Trigonometric Identities are the equalities that involve trigonometry functions and holds true for all the values of variables given in the equation.
With both x and y in the first quadrant, we have each of sin(x), cos(x), sin(y), and cos(y) being positive.
Then from the Pythagorean identity, it follows that
cos(x) = √(1 - sin²(x)) = 4/5
cos(y) = √(1 - sin²(y)) = √5/3
Using the angle sum identity for sine,
sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
[tex]\rm Sin(x+y)= (\dfrac{3}{5}) \times (\dfrac{\sqrt{5}}{3}) + (\dfrac{4}{5}) \times (\dfrac{2}{3})[/tex]
[tex]\rm Sin(x+y) = \dfrac{\sqrt{5}}{5} + \dfrac{8}{15}[/tex]
[tex]\rm Sin(x+y) = (\dfrac{3\sqrt{5})}{15} + \dfrac{8}{15}[/tex]
[tex]\rm Sin (x+y)=\dfrac{(3\sqrt{5} + 8)}{15}[/tex]
Hence the evaluation of sin(x+y) will be equal to sin(x+y) =(3√5 + 8)/15
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