Given that sin x=(1)/(sqrt26) and sin y=(1)/(sqrt5), and that angles x and y are both in Quadrant I, find the exact value of sin(x+y), in simplest radical form. Answer:

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Given that  sin x=(1)/(sqrt26) and  sin y=(1)/(sqrt5), and that angles  x and  y are both in Quadrant I, find the exact value of  sin(x+y), in simplest radical form. Answer:

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Use Sine Addition Formula: To find sin(x+y), we will use the sine addition formula: sin(x+y) = sin(x)cos(y) + cos(x)sin(y).Find cos(x) and cos(y): First, we need to find cos(x) and cos(y). Since x and y are in Quadrant I, both cos(x) and cos(y) will be positive. We can use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 to find cos(θ) for both angles.Calculate cos(x): For angle x, we have sin(x) = 1/√26. So, sin^2(x) = (1/√26)^2 = 1/26. Using the Pythagorean identity, we get cos^2(x) = 1 - sin^2(x) = 1 - 1/26 = 25/26. Therefore, cos(x) = √(25/26) = 5/√26.Calculate cos(y): For angle y, we have sin(y) = 1/√5. So, sin^2(y) = (1/√5)^2 = 1/5. Using the Pythagorean identity, we get cos^2(y) = 1 - sin^2(y) = 1 - 1/5 = 4/5. Therefore, cos(y) = √(4/5) = 2/√5.Substitute values into formula: Now we can substitute the values into the sine addition formula: sin(x+y) = sin(x)cos(y) + cos(x)sin(y) = (1/√26)(2/√5) + (5/√26)(1/√5).Simplify expression: Simplify the expression: sin(x+y) = (2/√130) + (5/√130) = (2+5)/√130 = 7/√130.Rationalize the denominator: To rationalize the denominator, we multiply the numerator and the denominator by √130: sin(x+y) = (7/√130) * (√130/√130) = 7√130/130.Final simplification: Finally, we simplify the expression: sin(x+y) = 7√130/130.

Given that $\sin x=\frac{1}{\sqrt{26}}$ and $\sin y=\frac{1}{\sqrt{5}}$ , and that angles $x$ and $y$ are both in Quadrant I, find the exact value of $\sin (x+y)$ , in simplest radical form. $\newline$ Answer:

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Given that $\sin x=\frac{1}{\sqrt{26}}$ and $\sin y=\frac{1}{\sqrt{5}}$ , and that angles $x$ and $y$ are both in Quadrant I, find the exact value of $\sin (x+y)$ , in simplest radical form. $\newline$ Answer: