First, let’s start by evaluating the given mathematical problem: [tex]\( y = \sqrt{\sin x - \cos x} \)[/tex] with [tex]\( x = 45^\circ \)[/tex].
Step 1: Convert the angle to radians.
Since trigonometric functions in mathematics often use radians, we'll convert [tex]\( 45^\circ \)[/tex] to radians. The conversion from degrees to radians is done using the formula:
[tex]\[ \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \][/tex]
For [tex]\( x = 45^\circ \)[/tex]:
[tex]\[ x = 45^\circ \times \frac{\pi}{180} = \frac{\pi}{4} \][/tex]
Step 2: Calculate [tex]\(\sin x\)[/tex] and [tex]\(\cos x\)[/tex] for [tex]\( x = \frac{\pi}{4} \)[/tex].
We know that:
[tex]\[ \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \][/tex]
[tex]\[ \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \][/tex]
Step 3: Substitute [tex]\(\sin x\)[/tex] and [tex]\(\cos x\)[/tex] into the expression [tex]\(\sin x - \cos x\)[/tex].
[tex]\[ \sin \left( \frac{\pi}{4} \right) - \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 0 \][/tex]
Step 4: Calculate the value of [tex]\( y \)[/tex].
[tex]\[ y = \sqrt{\sin x - \cos x} = \sqrt{0} = 0 \][/tex]
Conclusion:
For [tex]\( x = 45^\circ \)[/tex], the value of [tex]\( y \)[/tex] is [tex]\( y = 0 \)[/tex]. The provided angle [tex]\( y = 90^\circ \)[/tex] does not alter the evaluation of the expression since it's only involved in the problem statement to check the result of [tex]\( y = 0 \)[/tex]. Hence, the solution has been evaluated correctly without the angle [tex]\( y \)[/tex].
Thus, the final value is:
[tex]\[ y = 0 \][/tex]
