To solve for [tex]\(c\)[/tex] using the equation
[tex]\[ c^2 = 3^2 + 10^2 - 2 \cdot 3 \cdot 10 \cdot \cos(34^\circ), \][/tex]
we will follow these steps:
### Step 1: Calculate the Values of the Known Quantities
First, compute the values of the squares of sides [tex]\(a\)[/tex] and [tex]\(b\)[/tex], which are given by:
[tex]\[ a = 3 \quad \Rightarrow \quad a^2 = 3^2 = 9 \][/tex]
[tex]\[ b = 10 \quad \Rightarrow \quad b^2 = 10^2 = 100 \][/tex]
### Step 2: Convert the Angle to Radians
Since the cosine function in many calculator tools uses radians, convert [tex]\(34^\circ\)[/tex] to radians.
[tex]\[ \text{Angle in radians} \approx 0.5934 \][/tex]
### Step 3: Compute the Cosine of the Angle
Now, find the cosine of [tex]\(34^\circ\)[/tex] (approximately).
[tex]\[ \cos(34^\circ) \approx 0.8290 \][/tex]
### Step 4: Apply the Law of Cosines Formula
Insert the computed values into the law of cosines formula:
[tex]\[ c^2 = 9 + 100 - 2 \cdot 3 \cdot 10 \cdot 0.8290 \][/tex]
### Step 5: Simplify the Expression
First, calculate the product of [tex]\(2 \cdot 3 \cdot 10 \cdot 0.8290\)[/tex]:
[tex]\[ 2 \cdot 3 \cdot 10 \cdot 0.8290 \approx 49.74 \][/tex]
Next, substitute this value back into the equation:
[tex]\[ c^2 = 9 + 100 - 49.74 \][/tex]
[tex]\[ c^2 = 109 - 49.74 \][/tex]
[tex]\[ c^2 = 59.26 \][/tex]
### Step 6: Find the Square Root to Determine [tex]\(c\)[/tex]
Take the square root of both sides to solve for [tex]\(c\)[/tex]:
[tex]\[ c = \sqrt{59.26} \approx 7.698 \][/tex]
### Step 7: Round the Result to the Nearest Tenth
Finally, round the result to the nearest tenth:
[tex]\[ c \approx 7.7 \][/tex]
Therefore, the value of [tex]\(c\)[/tex], rounded to the nearest tenth, is [tex]\(7.7\)[/tex].
