The point (3,-8) is on the terminal arm of the angle x. Find sinx exactly and find x, in radians, to two decimal places.
I don't how I should start this question.
please help and thank you
you know that tanθ = y/x = -8/3
Arctan(8/3) = 1.21
so, since we are in the 4th quadrant,
θ = -1.21
To find sinx exactly, we can use the Pythagorean Identity. However, we need to first determine the coordinates of the point on the unit circle that corresponds to the angle x.
The given point (3,-8) is in the fourth quadrant of the Cartesian coordinate system. Since the x-coordinate is positive and the y-coordinate is negative, we can infer that the value of sinx is negative.
To find the length of the vector from the origin to the point (3,-8), we can use the distance formula, which is derived from the Pythagorean Theorem:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) represents the coordinates of the origin, (0,0), and (x2, y2) represents the coordinates of the given point (3,-8).
Applying the formula, we have:
d = sqrt((3 - 0)^2 + (-8 - 0)^2)
= sqrt(3^2 + (-8)^2)
= sqrt(9 + 64)
= sqrt(73)
Since the point (3,-8) lies on the terminal arm of angle x, the value of sinx can be found by dividing the y-coordinate by the length of the vector:
sinx = y / d
= -8 / sqrt(73)
To express sinx exactly, we rationalize the denominator:
sinx = (-8 / sqrt(73)) * (sqrt(73) / sqrt(73))
= -8sqrt(73) / 73
Therefore, sinx = -8sqrt(73) / 73.
To find x in radians to two decimal places, we can use the inverse sine function, sin^(-1):
x = sin^(-1)(sinx)
= sin^(-1)(-8sqrt(73) / 73)
Using a calculator, we can evaluate this expression to obtain the value of x in radians.