Understanding how to determine the angles of a rhombus using trigonometry when the side length and area are given is crucial in geometry. In this guide, we'll explain the process step-by-step, ensuring clarity and ease of understanding with actual numbers and calculations.
Step 1: Understand the Key Formulas
To determine the angles of a rhombus, we need to use the following formulas:
1. **Area of a rhombus**:
\[ A = a^2 \sin(\theta) \]
where \( a \) is the length of the side and \( \theta \) is one of the angles (either \( \angle A \) or \( \angle B \)).
2. **Sum of angles in a rhombus**:
\[ \angle A + \angle B = 180^\circ \]
Step 2: Explain the Formulas
1. The area formula \( A = a^2 \sin(\theta) \) relates the area of the rhombus to the side length and one of its angles. By rearranging this formula, we can solve for the angle.
2. The sum of angles formula \( \angle A + \angle B = 180^\circ \) helps us determine the other angle once we find one angle, since a rhombus is a type of parallelogram and opposite angles are equal.
Step 3: Use Actual Numbers as an Example
Let's assume the side length (\( a \)) is 10 units and the area (\( A \)) is 80 square units.
Step 4: Calculate Angle \( \theta \)
First, we need to find the angle \( \theta \) using the area formula. Rearrange the formula to solve for \( \theta \):
\[ \sin(\theta) = \frac{A}{a^2} \]
Substitute the known values:
\[ \sin(\theta) = \frac{80}{10^2} \]
\[ \sin(\theta) = \frac{80}{100} \]
\[ \sin(\theta) = 0.8 \]
Now, find \( \theta \) using the inverse sine function:
\[ \theta = \sin^{-1}(0.8) \]
Using a calculator:
\[ \theta \approx 53.13^\circ \]
So, \( \angle A \approx 53.13^\circ \).
Step 5: Calculate Angle \( B \)
Since \( \angle A + \angle B = 180^\circ \), we can find \( \angle B \):
\[ \angle B = 180^\circ - \angle A \]
Substitute the value of \( \angle A \):
\[ \angle B = 180^\circ - 53.13^\circ \]
\[ \angle B \approx 126.87^\circ \]
Final Values
- \( \angle A \approx 53.13^\circ \)
- \( \angle B \approx 126.87^\circ \)
By following these steps, you can easily determine the angles of a rhombus given the side length and area. Understanding these geometric principles will enhance your problem-solving skills and mathematical knowledge.
