Sure! Let's go through the solution step-by-step using the Law of Cosines.
Given:
- [tex]\( a = 5 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- [tex]\( C = 45^\circ \)[/tex] (the angle in degrees)
We need to find the length of side [tex]\( c \)[/tex], which is opposite the given angle [tex]\( C \)[/tex].
### Step 1: Convert angle [tex]\( C \)[/tex] to radians
The Law of Cosines requires the angle in radians. To convert degrees to radians, we use the formula:
[tex]\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \][/tex]
So,
[tex]\[ C_{\text{radians}} = 45^\circ \times \frac{\pi}{180} \approx 0.7854 \text{ radians} \][/tex]
### Step 2: Apply the Law of Cosines
The Law of Cosines formula is:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
Substituting the given values:
[tex]\[ a = 5 \][/tex]
[tex]\[ b = 7 \][/tex]
[tex]\[ \cos(C_{\text{radians}}) = \cos(0.7854) \][/tex]
We need the cosine of [tex]\( 0.7854 \)[/tex]. Using a calculator, we find:
[tex]\[ \cos(0.7854) \approx 0.7071 \][/tex]
Thus, plugging in the values:
[tex]\[ c^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot 0.7071 \][/tex]
Calculating each term:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ 7^2 = 49 \][/tex]
[tex]\[ 2 \cdot 5 \cdot 7 \cdot 0.7071 \approx 49.4975 \][/tex]
Putting it all together:
[tex]\[ c^2 = 25 + 49 - 49.4975 \][/tex]
[tex]\[ c^2 \approx 24.5025 \][/tex]
### Step 3: Solve for [tex]\( c \)[/tex]
Now, solve for [tex]\( c \)[/tex] by taking the square root of both sides:
[tex]\[ c = \sqrt{24.5025} \][/tex]
[tex]\[ c \approx 4.95 \][/tex]
### Step 4: Round to the nearest tenth
Finally, rounding [tex]\( 4.95 \)[/tex] to the nearest tenth:
[tex]\[ c \approx 5.0 \][/tex]
Therefore, the length of side [tex]\( c \)[/tex] is approximately [tex]\( 5.0 \)[/tex] units.
