Question

(50 points!!!!!)

Find the exact value of; arctan(sin (pi/2)).

please show your work using a unit circle.

Answer 1

π/4

Step-by-step explanation:

(I cannot attach a unit circle here, sorry :)

Recall the function of the (x,y) values of the unit circle. X represents the cosine value, and y is the sine value.

If you look at a unit circle, π/2 has the coordinates (0,1), so the sine value of π/2 is 1.

To find arctangent, remember that the range of tangent is -π/2 < x < π/2. In terms of the unit circle, these are the minimum and maximum values, and we cannot use -π/2 and π/2 since x is 0, which means it'll be undefined. The angle where the coordinates can be found is the arctangent of the function.

So, find the coordinate where both x and y have the same value (√2/2, √2/2) at π/4. Therefore, the answer for the arctan(sin(π/2)) = π/4.

Answer 2

The exact value of arctan(sin (pi/2)) is pi/4. This was determined by first finding the value of sin(pi/2) on the unit circle, which is 1, and then finding the angle whose tangent is 1, which is at pi/4.

Step-by-step explanation:

You are asked to find the exact value of arctan(sin (pi/2)) using a unit circle. First, let's find the value of sin(pi/2). On the unit circle, sin represents the y-coordinate of a point. At an angle of pi/2 (or 90 degrees), we are at the top of the circle where the y-coordinate is 1. So, sin(pi/2) is 1.

Next, we need to find the angle whose tangent is 1. Remember that tangent is the ratio of the y-coordinate to the x-coordinate (sin/cos), and it equals 1 when these two coordinates are the same. This occurs at pi/4 (or 45 degrees), where the coordinates are (1/root 2, 1/root 2).

Therefore, arctan(sin (pi/2)) is pi/4, because the tangent of pi/4 is 1, the same as sin (pi/2).