Problem 54 - Precalculus Mathematics for Calculus

Question

Find the area of a triangle with sides of length 7 and 9 and included angle \(72^{\circ}\).

Step-by-Step Solution

Answer

The area of the triangle is approximately 29.96 square units.

Step 1: Understand the Triangle Area Formula

For a triangle with two sides of lengths \(a\) and \(b\) and the included angle \(C\), the area \(A\) can be found using the formula: \[ A = \frac{1}{2}ab \sin(C) \] In this problem, the lengths \(a\) and \(b\) are 7 and 9, and the included angle \(C\) is \(72^{\circ}\).

Step 2: Substitute Values into Formula

Substitute the known values into the area formula: \[ A = \frac{1}{2} \times 7 \times 9 \times \sin(72^{\circ}) \] This simplifies to: \[ A = \frac{1}{2} \times 63 \times \sin(72^{\circ}) \]

Step 3: Calculate the Sine of the Angle

Calculate \(\sin(72^{\circ})\). Using a calculator, \(\sin(72^{\circ}) \approx 0.9511\).

Step 4: Calculate the Area

Substitute \(\sin(72^{\circ})\) into the formula: \[ A = \frac{1}{2} \times 63 \times 0.9511 \] Calculate this to find the area: \[ A \approx \frac{1}{2} \times 63 \times 0.9511 \approx 29.96265 \]

Step 5: Round the Area

Round the result to a reasonable number of decimal places. In this case, the area of the triangle is approximately \(29.96\).

Key Concepts

Triangle Area Formula

In trigonometry, finding the area of a triangle becomes a breeze once you understand the formula involving two sides and the sine of their included angle. Imagine having two sides of a triangle, labelled as \(a\) and \(b\), with an angle \(C\) between them. The formula to compute the area \(A\) is:

• \[ A = \frac{1}{2}ab \sin(C) \]

This formula is incredibly useful because it allows you to calculate the area without needing to know the height of the triangle. You simply need the lengths of two sides and the measure of the angle which sits between them.
This method of calculating area is especially applicable when the height is not easily obtainable or when dealing with non-right triangular shapes.
Embracing this formula gives you more flexibility in solving geometric problems.

Sine Function

The sine function is a fundamental concept in trigonometry, helping to relate angles to side lengths in right-angled triangles. It is defined for an angle in a right triangle as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

• Mathematically, \( \sin(C) = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \).

However, its use is not limited to right triangles, thanks to its extension through the unit circle which allows \( \sin \) to operate for any angle. When you use a calculator to find \( \sin(72^{\circ}) \), it uses these principles to provide an approximate value, around 0.9511.
This approximation is crucial when employing the triangle area formula, as it converts the angular information into a numerical form that describes the triangle’s geometric properties.

Included Angle

The included angle is a critical concept in both geometry and trigonometry. It's the angle formed between two given sides of a triangle. In the scenario of the triangle area formula, this angle is pivotal as it directly impacts the calculated area.

• The greater the included angle, the larger the value of \( \sin(C) \) might be, thus potentially increasing the area.
• Conversely, a smaller angle often results in a smaller sine value, which could lead to a decreased area.

The included angle's role in determining area highlights the interconnectedness of trigonometric concepts, where angles not only define shape but dictate size. Understanding this angle is fundamental when applying formulas that intertwine linear dimensions and angular measures.
Thus, knowing how to correctly identify and utilize the included angle is critical when working with non-right-angled triangles.