the point P(-5,-3) lies on the terminal arm of an angle theta in standard position. what is the measure of theta?
To find the measure of angle theta, we can use the trigonometric functions sine and cosine.
We know that the point P(-5, -3) lies on the terminal arm of the angle theta. From this information, we can determine the values of the sine and cosine of theta.
The cosine of an angle in standard position is calculated as the x-coordinate of the point on the unit circle corresponding to that angle. In this case, the x-coordinate is -5.
So, cos(theta) = -5.
The sine of an angle in standard position is calculated as the y-coordinate of the point on the unit circle corresponding to that angle. In this case, the y-coordinate is -3.
So, sin(theta) = -3.
Next, let's find the measure of angle theta by using the inverse trigonometric functions.
Using the information we have, we can calculate the value of theta using the equation:
theta = arctan(-3/-5) = arctan(3/5).
Taking the arctan of 3/5, we find that theta is approximately 30.96 degrees.
Therefore, the measure of theta is approximately 30.96 degrees.
To find the measure of theta, we can use the concept of trigonometry and the coordinates of the point P(-5,-3) on the terminal arm.
In standard position, the positive x-axis lies to the right and the positive y-axis lies upwards. The terminal arm of an angle is the arm that rotates to the position of the point P.
First, we can find the length of the line segment from the origin (0, 0) to the point P(-5, -3) using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-5 - 0)^2 + (-3 - 0)^2)
= sqrt((-5)^2 + (-3)^2)
= sqrt(25 + 9)
= sqrt(34)
The length of the line segment from the origin to point P is sqrt(34).
Now, we can use the trigonometric ratio of the coordinates of point P to find the measure of theta.
sin(theta) = opposite / hypotenuse
sin(theta) = (-3) / sqrt(34)
To find theta, we can take the inverse sine (arcsine) of both sides:
theta = arcsin((-3) / sqrt(34))
Using a calculator, the approximate value of theta is -0.6476 radians or -37.1195 degrees (rounded to four decimal places).
Therefore, the measure of theta is approximately -0.6476 radians or -37.1195 degrees.
tanθ = y/x = 0.6
But you are in QIII, so add π to that reference angle.