Assume that alpha is an angle in standard position whose terminal side contains the given point and that 0 < alpha < pi/2. Find the radian measure of alpha to the nearest 10th
Since 0 < alpha < pi/2, alpha is an acute angle.
To find the radian measure of alpha, we need to find the length of the arc intercepted by the angle alpha on the unit circle.
Since alpha is an acute angle, the terminal side of alpha will intersect the unit circle at a point on the first quadrant.
If we draw a line from the origin to the point of intersection, we have a right triangle formed. Let's call the hypotenuse of this triangle r (which is the radius of the unit circle) and the length of the adjacent side x.
Since the hypotenuse of a right triangle in the unit circle is 1, we have:
r = 1
By the definition of cosine, we have:
cos(alpha) = adjacent/hypotenuse = x/r = x/1 = x
So, cos(alpha) = x
Since we are given a specific point, we can use the coordinates of the point to find x.
Let's assume that the given point is (a, b), where a is the x-coordinate and b is the y-coordinate.
Since the point lies on the unit circle, the coordinates satisfy the equation for the unit circle: x^2 + y^2 = 1
Substituting the given point (a, b) into the equation, we have:
a^2 + b^2 = 1
Solving for x, we have:
x = a
Since cos(alpha) = x, we have:
cos(alpha) = a
To find the radian measure of alpha, we need to find the angle alpha that has a cosine value of a.
Using the inverse cosine function, we have:
alpha = arccos(a)
Using a calculator, we can find the approximate value of arccos(a) in radians.
Hence, the radian measure of alpha to the nearest tenth is arccos(a), where a is the x-coordinate of the given point.