Find the exact value:
1.) cos^-1(sin(pi/2))
2.) arccos(cos(7pi/2))
To find the exact values of the given trigonometric expressions, we need to use the definitions of inverse trigonometric functions and some trigonometric properties.
1.) Let's find the value of cos^(-1)(sin(pi/2)):
First, we need to evaluate sin(pi/2). Since sin(pi/2) is the sine of the angle pi/2, which is equal to 90 degrees, we know that sin(pi/2) = 1.
Now, we can substitute this value into the expression cos^(-1)(sin(pi/2)). Applying the definition of the inverse cosine function, cos^(-1)(1) = pi/2.
Therefore, cos^(-1)(sin(pi/2)) = pi/2.
2.) Let's find the value of arccos(cos(7pi/2)):
First, we need to evaluate cos(7pi/2). Since cos(7pi/2) is the cosine of the angle 7pi/2, we know that this angle is equivalent to 3.5 complete revolutions or 7 * 360 degrees + 180 degrees = 2520 degrees + 180 degrees = 2700 degrees.
Now, we can convert the angle back to radians by dividing 2700 degrees by 180 degrees/pi, which is pi/2 radians per degree: 2700 degrees * (pi/2 radians/1 degree) = (2700/180) * pi/2 = 15pi radians.
Next, we apply the definition of the inverse cosine function, which returns the angle whose cosine is equal to the given value. Since cos(7pi/2) = cos(15pi), we need to find the angle theta such that cos(theta) = cos(15pi).
Using the periodicity property of the cosine function, we know that cos(theta) = cos(theta + 2pi*n), where n is an integer. So, we can subtract any multiple of 2pi from the given angle to find the equivalent angle within the domain of the inverse cosine function.
In this case, we subtract 2pi from 15pi to get an equivalent angle within the range [-pi, pi]:
15pi - 2pi = 13pi.
Therefore, arccos(cos(7pi/2)) = 13pi.