Given that cos x=(4)/(5) and sin y=(sqrt8)/(3), and that angles x and y are both in Quadrant I, find the exact value of sin(x+y), in simplest radical form. Answer:

Question

Given that  cos x=(4)/(5) and  sin y=(sqrt8)/(3), and that angles  x and  y are both in Quadrant I, find the exact value of  sin(x+y), in simplest radical form. Answer:

Bytelearn · Accepted Answer

Apply Sine Addition Formula: Use the sine addition formula: sin(x+y) = sin(x)cos(y) + cos(x)sin(y). First, we need to find sin(x) and cos(y). Since cos x = 4/5 and x is in Quadrant I, we can find sin(x) using the Pythagorean identity sin^2(x) + cos^2(x) = 1.Calculate sin(x): Calculate sin(x): sin^2(x) = 1 - cos^2(x) sin^2(x) = 1 - (4/5)^2 sin^2(x) = 1 - 16/25 sin^2(x) = (25/25) - (16/25) sin^2(x) = 9/25 sin(x) = sqrt(9/25) sin(x) = 3/5 Since x is in Quadrant I, sin(x) is positive.Find cos(y): Now, find cos(y): We are given sin y = sqrt(8)/3. Using the Pythagorean identity again, cos^2(y) + sin^2(y) = 1. cos^2(y) = 1 - sin^2(y) cos^2(y) = 1 - (sqrt(8)/3)^2 cos^2(y) = 1 - 8/9 cos^2(y) = (9/9) - (8/9) cos^2(y) = 1/9 cos(y) = sqrt(1/9) cos(y) = 1/3 Since y is in Quadrant I, cos(y) is positive.Use Sine Addition Formula: Now that we have sin(x) = 3/5 and cos(y) = 1/3, we can use the sine addition formula: sin(x+y) = sin(x)cos(y) + cos(x)sin(y) sin(x+y) = (3/5)(1/3) + (4/5)(sqrt(8)/3) sin(x+y) = 3/15 + 4sqrt(8)/15 sin(x+y) = 1/5 + 4sqrt(8)/15Combine Terms: Combine the terms to get the final answer: sin(x+y) = (1/5)(3/3) + 4sqrt(8)/15 sin(x+y) = 3/15 + 4sqrt(8)/15 sin(x+y) = (3 + 4sqrt(8))/15

Given that $\cos x=\frac{4}{5}$ and $\sin y=\frac{\sqrt{8}}{3}$ , 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:

Full solution

More problems from Find trigonometric ratios using multiple identities

Related topics

Given that cos⁡x=45 \cos x=\frac{4}{5} cosx=54​ and sin⁡y=83 \sin y=\frac{\sqrt{8}}{3} siny=38​​, and that angles x x x and y y y are both in Quadrant I, find the exact value of sin⁡(x+y) \sin (x+y) sin(x+y), in simplest radical form.\newlineAnswer:

Given that $\cos x=\frac{4}{5}$ and $\sin y=\frac{\sqrt{8}}{3}$ , 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: