The point P(k, 24) is 25 units from the origin. If P lies on the terminal arm of an angle, theta, in standard position, 0 < theta < 360,determine the measure of theta.
Unsure how to go on from here. I'd appreciate if someone can guide me through the first few steps
Thank you
sinTheta=24/25
theta=arcsin(24/25)
Thank you! can you check my answer please?
I get 74 degrees and 106 degrees
in round numbers yes. I would have taken the asnwers to nearest tenth of degrees.
Thank you again! We have round to the nearest degree unless specified =)
To determine the measure of theta, we need to use the distance formula in the coordinate plane. The distance between two points (x1, y1) and (x2, y2) is given by the formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, we know the point P(k, 24) is 25 units away from the origin (0, 0).
So, we can set up the distance formula:
25 = sqrt((k - 0)^2 + (24 - 0)^2)
Simplifying the equation, we have:
625 = k^2 + 576
Rearranging the terms:
k^2 = 625 - 576
k^2 = 49
Taking the square root of both sides:
k = ±7
Since 0 < theta < 360, we need to consider the positive value of k, which is k = 7.
Now, we know the x-coordinate of the point P is 7, and the y-coordinate is 24. To find the angle theta, we can use inverse trigonometric functions.
We have:
tan(theta) = y/x
tan(theta) = 24/7
Now, we need to find the angle whose tangent is 24/7. We can use the inverse tangent function (also known as arctan or tan^(-1)).
theta = arctan(24/7)
Using a calculator or a trigonometric table, we can find the approximate value of theta:
theta ≈ 74.38 degrees
Therefore, the measure of theta is approximately 74.38 degrees.