Certainly! Let's go through the problem step by step. We want to evaluate the expression:
[tex]\[ 4^2 + 5^2 - 2 \cdot 4 \cdot 5 \cos(C) \][/tex]
where [tex]\( C = 2^2 \)[/tex].
### Step 1: Calculate [tex]\( 4^2 \)[/tex]
[tex]\[ 4^2 = 16 \][/tex]
### Step 2: Calculate [tex]\( 5^2 \)[/tex]
[tex]\[ 5^2 = 25 \][/tex]
### Step 3: Calculate [tex]\( C \)[/tex]
Since [tex]\( C = 2^2 \)[/tex],
[tex]\[ C = 4 \][/tex]
### Step 4: Calculate [tex]\(\cos(C)\)[/tex]
Here, we need the value of [tex]\(\cos(4)\)[/tex].
### Step 5: Evaluate the term [tex]\( -2 \cdot 4 \cdot 5 \cos(4) \)[/tex]
Let's break it down:
[tex]\[ 2 \cdot 4 \cdot 5 = 40 \][/tex]
So the term becomes:
[tex]\[ -40 \cos(4) \][/tex]
### Step 6: Plug in the value of [tex]\(\cos(4)\)[/tex]
Given:
[tex]\[ \cos(4) \approx -0.6536436208636119 \][/tex]
Therefore:
[tex]\[ -40 \cos(4) = 40 \times (-0.6536436208636119) = -26.145744834544477 \][/tex]
### Step 7: Combine all terms
[tex]\[ 4^2 + 5^2 - 2 \cdot 4 \cdot 5 \cos(4) = 16 + 25 + (-26.145744834544477) \][/tex]
So, the final expression adds up to:
[tex]\[ 16 + 25 - 26.145744834544477 = 67.14574483454447 \][/tex]
### Conclusion
So, the detailed breakdown provides us:
1. [tex]\(4^2 = 16\)[/tex]
2. [tex]\(5^2 = 25\)[/tex]
3. [tex]\( -2 \cdot 4 \cdot 5 \cos(4) = -26.145744834544477 \)[/tex]
4. Resulting in an expression value of [tex]\(67.14574483454447\)[/tex]
Thus, the final answer to the original question is:
[tex]\[67.14574483454447\][/tex]
