Problem 54 - Algebra and Trigonometry

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 29.98 square units.

Step 1: Formula for Triangle Area

The area of a triangle when two sides and the included angle are known can be calculated using the formula: \[ A = \frac{1}{2}ab\sin(C) \] where \(a\) and \(b\) are the lengths of the sides, and \(C\) is the included angle in degrees.

Step 2: Substitute Known Values

Substitute the known values into the formula. In this case, \(a = 7\), \(b = 9\), and \(C = 72^{\circ}\). So the formula becomes:\[ A = \frac{1}{2} \times 7 \times 9 \times \sin(72^{\circ}) \]

Step 3: Calculate Sine of the Angle

Use a calculator to find \(\sin(72^{\circ})\). The sine of 72 degrees is approximately 0.9511.

Step 4: Calculate the Area

Substitute \(\sin(72^{\circ})\) into the formula to get:\[ A = \frac{1}{2} \times 7 \times 9 \times 0.9511 \]Calculate to find the area.\[ A \approx \frac{1}{2} \times 63 \times 0.9511 = 29.98415 \]

Step 5: Final Result

The area of the triangle is approximately 29.98 square units.

Key Concepts

Triangle Area Formula

When you know two sides of a triangle and the angle between them, the area can be calculated easily. The formula for the area is useful in many geometric problems. It is written as:\[ A = \frac{1}{2}ab\sin(C) \]where:

• \(a\) and \(b\) are the lengths of the sides you know.
• \(C\) is the included angle, the angle formed between the two sides.

The reason this formula works is due to the properties of the sine function, which helps measure the height of the triangle indirectly through the angle. This method is especially helpful for non-right triangles, where traditional base-height formulas may be difficult to apply directly.This formula gives you a quick and clear way to determine the triangle's area without having to measure every little part of it, especially when an included angle makes direct measurement tough.

Sine Function

The sine function is one of the primary trigonometric functions often used in calculations involving triangles, cycles, and oscillations. In this specific context, the sine function is used to relate an angle in a triangle to the ratio of the length of the opposite side over the hypotenuse in a right triangle. However, even in non-right triangles, the sine function is very useful. It helps extend our calculations to situations outside simple right triangle conditions.For an angle \(\theta\), sine is defined in a unit circle as:\( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \)The usage of the sine function allows you to account for how much a triangle "opens". When you multiply \(\sin(\theta)\) by the sides, it effectively scales the line segment needed to complete the triangle's area calculation. In typical cases like the inclusion of non-right triangle angle measurements, tools like calculators help to quickly retrieve the value of \(\sin(\theta)\) for precise calculations.

Included Angle

The included angle in a triangle is specifically the angle between two known sides. It is crucial for applying various strategies, like the law of cosines or the triangle area formula. The angle directly "includes" those two sides in its measurement.In practice, knowing the included angle allows us to accurately calculate the area of a triangle using the formula:\[ A = \frac{1}{2}ab\sin(C) \]As this setup relies on accurate angle measurements, the included angle is often given or measured precisely, as a small error can significantly alter the final solution. The included angle provides a necessary pivot point for converting line segments into a calculable area. Therefore, recognizing and accurately measuring the included angle is essential to executing the formula correctly, ensuring precision in any geometric computation involving triangles.