To determine the value of [tex]\(\cos 45^\circ\)[/tex], we will follow through the steps and confirm which answer choice is correct.
### Step 1: Understanding the Angle and the Unit Circle
The cosine of an angle in a right triangle relates the adjacent side to the hypotenuse. For [tex]\(\cos 45^\circ\)[/tex], we will use the unit circle where [tex]\(45^\circ\)[/tex] is a standard angle.
### Step 2: Converting to Radians (Optional)
In trigonometry, [tex]\(45^\circ\)[/tex] can be represented in radians as:
[tex]\[ 45^\circ = \frac{\pi}{4} \][/tex]
### Step 3: Trigonometric Value of [tex]\(\cos 45^\circ\)[/tex]
The cosine of [tex]\(45^\circ\)[/tex] is a well-known value and can be derived from an isosceles right triangle (45-45-90 triangle) with equal legs:
[tex]\[ \cos 45^\circ = \cos \left( \frac{\pi}{4} \right) \][/tex]
### Step 4: Standard Value
In an isosceles right triangle, if the legs are of length 1, the hypotenuse will be [tex]\(\sqrt{2}\)[/tex]. Hence,
[tex]\[ \cos 45^\circ = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}} \][/tex]
### Step 5: Confirming with Multiple Choice Options
From the given options:
A. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
B. [tex]\(\sqrt{2}\)[/tex]
C. [tex]\(\frac{1}{2}\)[/tex]
D. 1
The option A, [tex]\(\frac{1}{\sqrt{2}}\)[/tex], matches our calculated value.
### Step 6: Verifying the Decimal Value (Optional)
The value of [tex]\(\frac{1}{\sqrt{2}}\)[/tex] can be approximated as:
[tex]\[ \frac{1}{\sqrt{2}} \approx 0.7071067811865476 \][/tex]
### Conclusion
Hence, the correct answer for [tex]\(\cos 45^\circ\)[/tex] is:
[tex]\[ \boxed{\frac{1}{\sqrt{2}}} \][/tex]
Therefore, the right choice among the given options is:
[tex]\[ \boxed{A} \][/tex]
