i really dont get this please help

Question

Find the exact value of arctan (sin (PI/2 )). For full credit, explain your reasoning.

Answer 1

Problem statement: Find the exact value of arctan( sin (PI/2) ).

First, notice that the angle Pi/2 is in units of radians, not degrees. 2*Pi radians = 360 degrees.

Pi/2 radians = 90 degrees.

Second, review the definition of the functions f(u) = sin(u) and g(v) = arctan(v). I am using u and v as variable names so that they don't get confused with the coordinates x and y in the Cartesian coordinate system.

sin(u) = the length of the side of a right triangle opposite the angle u, where 1 is the length of the hypotenuse of that right triangle. For angles of u > 90 degrees but less than 270 degrees, sin(u) will be negative. For angles of u > 270 degrees but less than 360 degrees, u will be positive.

This is because the hypotenuse of the right triangle used to calculate sin(u) traces out a unit circle going counter-clockwise around the origin of the Cartesian coordinate system. The (x,y) points on this unit circle = (cos(u),sin(u)). The angle u is measured counter-clockwise starting at 0 for the positive x-axis.

When u = Pi/2, the point (x,y) on the unit circle = (0,1). So sin(Pi/2) = sin(90 degrees) = 1.

arctan(v) = the size of the angle g in a right triangle, where the ratio of the length of the side opposite angle g (this length = the y coordinate of the point on a unit circle for angle g) divided by the length of the side adjacent to angle g (= the x coordinate of this point on the unit circle) = v. Because there are two points on the unit circle that can have the same ratio y/x, the range of the function arctan(v) is limited to -Pi/2 < arctan(v) < Pi/2.

When v =1, this means y/x = 1 for (x,y) coordinates on the unit circle. This occurs at the point (sqrt(2)/2,sqrt(2)/2) . Notice that (-sqrt(2)/2,-sqrt(2)/2) is out of range.

The angle g(1) = arctan(1) = arctan( sin(Pi/2) ) = 45 degrees = Pi/4

Answer 2

Hi, Ashley.

To find the value of arctan(sin(pi/2)), you have to evaluate two functions: First you find the value of sin(pi/2), and then you find the value of arctan(x) where x is the value of sin(pi/2).

I'm not sure how much trigonometry you've studied up to this point, but I will base my explanation on using angles of rotation and the unit circle. When the terminal side of an angle intersects the unit circle at a point (x, y), then x is the cosine of the angle and y is the sine of the angle. (Also, the ratio y/x is the tangent of the angle.) Since pi/2 represents an angle of rotation whose terminal side passes through the point (0, 1) on the unit circle, the y-coordinate of that point is the value of sin(pi/2). So, sin(pi/2) = 1.

You now need to determine what arctan(1) is. "Arctan" is a word used to describe the inverse tangent function. The arctan function accepts a real number as an input and outputs the angle of rotation for which the tangent of that angle is the real number. (Note that because the tangent function is periodic, there are infinitely many angles having the same tangent value. However, if we limit ourselves to angles between -pi/2 and pi/2, there is only one angle that has a particular tangent value.) In this case, you're finding the angle of rotation between -pi/2 and pi/2 that has a tangent value of 1. This happens at the point on the unit circle where the x- and y-coordinates are equal. That point is the point in the first quadrant where the line y = x intersects the unit circle, and the coordinates of that point are (sqrt(2)/2, sqrt(2)/2). The angle of rotation between -pi/2 and pi/2 associated with that point is pi/4.

So, arctan(sin(pi/2)) = pi/4.

Doug