To solve for $b$, we use the formula for the area of a triangle given two sides and the included angle:

$\text{Area} = \frac{1}{2} a b \sin C$

Given:

• $\text{Area} = 528.3$ mm²
• $a = 21$ mm
• $\angle C = 57^\circ$

Substituting these values:

$528.3 = \frac{1}{2} (21) b \sin(57^\circ)$

Solving for $b$:

$b = \frac{2 \times 528.3}{21 \times \sin(57^\circ)}$

Let's calculate the value. math

Given values

area = 528.3 # mm² a = 21 # mm angle_C = 57 # degrees

Convert angle to radians for sine function

angle_C_rad = math.radians(angle_C)

Solve for b

b = (2 * area) / (a * math.sin(angle_C_rad)) b_rounded = round(b, 1)

b_roundedThe length of $b$ is 60.0 mm when rounded to the nearest tenth.

Would you like a step-by-step breakdown of the calculations?

Related Questions:

1. How do you derive the triangle area formula using sine?
2. What happens if the given angle is obtuse instead of acute?
3. How can the Law of Sines be used to find other missing sides?
4. What is the significance of the sine function in triangle geometry?
5. How can you determine the height of the triangle using trigonometry?

Tip:

Always ensure your calculator is set to the correct mode (degrees or radians) when using trigonometric functions!