the point P(4,-3) lies on the terminal arm of an angle theta in standard position. determine the measure f theta to the nearest degree
Oh, good ol' theta! Let's figure out its measure, shall we?
First, we need to find the angle formed by the x-axis and the line passing through the origin and point P. This will be our reference angle.
Using some good ol' math, we can calculate the reference angle as such:
reference angle = atan(abs(y-coordinate of P / x-coordinate of P)
= atan(abs(-3 / 4))
= atan(0.75)
Now, let's use our trusty calculator to determine the value of theta in degrees. *Inserts clown nose*
Theta ≈ atan(0.75) ≈ 36.87 degrees
So, to the nearest degree, the measure of theta is approximately 37 degrees.
*Honks clown horn*
To determine the measure of angle theta in standard position, we can use the coordinates of point P(4, -3).
We can use the tangent function to find the angle's measure:
tan(theta) = y / x
Substituting the given values in the equation, we get:
tan(theta) = -3 / 4
Next, we can use the inverse tangent (arctan) to find the angle itself:
theta = arctan(-3 / 4)
Using a scientific calculator to find the arctan value, we get:
theta ≈ -36.87 degrees
Therefore, the measure of angle theta to the nearest degree is approximately -37 degrees.
To determine the measure of angle theta to the nearest degree, we can use trigonometry. Since the point P(4,-3) lies on the terminal arm of angle theta in standard position, we can form a right triangle where the adjacent side is 4 and the opposite side is -3.
Using the Pythagorean theorem, we can find the hypotenuse (r) of the triangle:
r^2 = (adjacent side)^2 + (opposite side)^2
r^2 = 4^2 + (-3)^2
r^2 = 16 + 9
r^2 = 25
r = 5
Now, we can use the ratio of the sides of the triangle to determine the trigonometric functions of angle theta:
sin(theta) = (opposite side) / (hypotenuse) = -3 / 5
cos(theta) = (adjacent side) / (hypotenuse) = 4 / 5
To find the measure of theta, we can use the inverse trigonometric functions. Since sin(theta) is negative and cos(theta) is positive in the third quadrant, theta must be in the third quadrant. Therefore, we need to find the angle whose sine is -3/5 in the third quadrant.
Using the inverse sine function, we can find the measure of theta:
theta = arcsin(-3/5) = -36.87 degrees
So, the measure of theta to the nearest degree is -37 degrees.
arctan(3/4) ≈ 37°
P is in QIV, so
θ = -37°
or
θ = 360-37 = 323°