Explain
cos^-1 [cos 5pi/4]
also tell it can be defined or not.
first let' s do
cos 5π/4 or cos 225°
Make a sketch of your right - angled isosceles triangle in quadrant III
we know cos 225 = -1/√2
so
cos^-1 [cos (5π/4) }
= cos^-1 [ -1/√2)
= 5π/4
or
= 3π/4 --- in the 2nd quadrant, by CAST the cosine is negative in II or III
try this on your calculator:
(set to radians)
enter:
cos
(5xπ÷4) =
--- you should see : -.707...
now press:
2ndF cos
=
--- you should see 2.356... which is the value of 3π/4
(your calculator is programmed to always give you the closest angle to zero, 3π/4 (135°) is closer to 0 than 5π/4 (225°) )
in general , for any mathematical operator 'job'
job^-1 (job (k) ) = k
To explain cos^-1 [cos (5π/4), we first need to understand the concept of inverse cosine or arccosine.
The arccosine (cos^-1) is the inverse function of the cosine (cos) function. It takes a value between -1 and 1 and returns the angle whose cosine value is equal to the given input.
In this case, we have cos^-1 [cos (5π/4)]. The argument of the cosine function is 5π/4.
To solve this, we need to determine if 5π/4 is within the range of the domain of the inverse cosine function.
The domain of the inverse cosine function is [-1, 1], meaning it can only take values between -1 and 1. If the argument of the cosine function is outside this range, the answer will be undefined.
Now, let's determine whether 5π/4 is within the domain of the inverse cosine function.
The cosine function oscillates between -1 and 1 within one period, which is 2π. So, any angle that differs from the reference angle by an integer multiple of 2π will have the same cosine value.
To find the equivalent cosine in the range [-1, 1] for 5π/4, we subtract or add multiples of 2π until we get an angle within the range.
5π/4 - 2π = -3π/4 (not within the range)
5π/4 - 4π = -3π/4 (not within the range)
5π/4 - 6π = -7π/4 (not within the range)
5π/4 + 2π = 13π/4 (not within the range)
5π/4 + 4π = 21π/4 (not within the range)
5π/4 + 6π = 29π/4 (not within the range)
After trying different combinations of adding or subtracting multiples of 2π, we see that we cannot find an angle that falls within the range [-1, 1] for the cosine function. Therefore, cos^-1 [cos (5π/4)] is undefined.
In summary, cos^-1 [cos (5π/4)] is not defined because the argument of the cosine function, 5π/4, is outside the range [-1, 1].