To find the value of alpha within the given range, we first need to determine the reference angle for alpha by finding the angle in the second quadrant that has a sine of 3/5. We can use the Pythagorean theorem to find the third side of the right triangle.
sin(alpha) = opposite/hypotenuse = -3/5 Let opposite = -3k and hypotenuse = 5k for some positive value of k
By the Pythagorean theorem: (-3k)^2 + b^2 = (5k)^2 9k^2 + b^2 = 25k^2 b^2 = 16k^2 b = 4k
So, the triangle in the second quadrant with a sine of 3/5 has sides of -3, 4, and 5. Therefore, the reference angle for alpha is arcsin(3/5) which is approximately 0.6435 radians.
Since alpha is in the third or fourth quadrant (where sine is negative), we have: alpha = pi - 0.6435 alpha = 2.497 radians
Therefore, sin(alpha) = sin(2.497) = -3/5 as given.
sin(alpha) = -3/5, 3pi/2 < alpha < 2pi
To find the value of alpha within the given range, we first need to determine the reference angle for alpha by finding the angle in the second quadrant that has a sine of 3/5. We can use the Pythagorean theorem to find the third side of the right triangle.
sin(alpha) = opposite/hypotenuse = -3/5
Let opposite = -3k and hypotenuse = 5k for some positive value of k
By the Pythagorean theorem:
(-3k)^2 + b^2 = (5k)^2
9k^2 + b^2 = 25k^2
b^2 = 16k^2
b = 4k
So, the triangle in the second quadrant with a sine of 3/5 has sides of -3, 4, and 5. Therefore, the reference angle for alpha is arcsin(3/5) which is approximately 0.6435 radians.
Since alpha is in the third or fourth quadrant (where sine is negative), we have:
alpha = pi - 0.6435
alpha = 2.497 radians
Therefore, sin(alpha) = sin(2.497) = -3/5 as given.