Problem 44 - Algebra and Trigonometry with Analytic Geometry

Question

Approximate the area of a parallelogram that has sides of lengths \(a\) and \(b\) (in feet) if one angle at a vertex has measure \(\boldsymbol{\theta}\). $$a=40.3, \quad b=52.6, \quad \theta=100^{\circ}$$

Step-by-Step Solution

Answer

The approximated area of the parallelogram is 2086.6 square feet.

Step 1: Convert Angle to Radians

To use the sine function in calculations, it's often easier to convert the angle from degrees to radians. The angle \( \theta = 100^{\circ} \) can be converted to radians using the formula \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \). So, \( \theta_{radians} = 100 \times \frac{\pi}{180} \approx 1.745 \text{ radians}\).

Step 2: Use Formula for Area of a Parallelogram

The area \(A\) of a parallelogram is given by \(A = ab \sin(\theta)\), where \(a\) and \(b\) are the lengths of the sides and \(\theta\) is the angle between them. Thus, we calculate \(A = 40.3 \times 52.6 \times \sin(1.745)\).

Step 3: Calculate Sine of Angle

Find \(\sin(1.745)\) using a calculator. This gives us approximately \(\sin(1.745) \approx 0.9848\).

Step 4: Calculate the Area

Substitute the values to find the area: \(A = 40.3 \times 52.6 \times 0.9848 = 2086.6 \text{ square feet (approximate)}\).

Key Concepts

Area of Parallelogram

The area of a parallelogram can be found using a simple yet powerful formula. To calculate it, you need two side lengths, say \(a\) and \(b\), and the angle \(\theta\) between them. The formula is \(A = ab \sin(\theta)\). This formula comes from the fact that a parallelogram can be divided into two congruent triangles. The base of these triangles is one side of the parallelogram, and the height is calculated from the other side and the angle between them. Thus, using trigonometry, the height is \(b \sin(\theta)\). Therefore, multiplying the base \(a\) by the height \(b \sin(\theta)\) gives you the area.

Trigonometric Functions

Trigonometric functions are essential tools in geometry for dealing with angles and sides of triangles. The main functions are sine, cosine, and tangent. Each of these functions relates the angles inside a right triangle to the ratios of its sides. The sine function, for instance, is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. Trigonometry not only helps in calculating lengths and angles in triangles but also extends to solving problems in circles and parallelograms, like calculating areas based on angles.

• Sine (\(\sin\)): opposite/hypotenuse
• Cosine (\(\cos\)): adjacent/hypotenuse
• Tangent (\(\tan\)): opposite/adjacent

Understanding these functions is crucial for solving many problems in mathematics and physics.

Radian Conversion

Converting degrees to radians is a fundamental skill in trigonometry. It is necessary because many math calculations, especially those involving trigonometric functions, are simpler in radians. The conversion is straightforward: multiply the degree measure by \(\pi/180\). This works because 180 degrees is equivalent to \(\pi\) radians.For example, if you have an angle of \(100^{\circ}\), you convert it to radians as follows:\[\theta_{radians} = 100 \times \frac{\pi}{180} \approx 1.745 \]This conversion is especially useful in trigonometry and calculus, where radian measures often make calculations more straightforward.

Sine Function

The sine function is a fundamental trigonometric function that connects an angle to a ratio of sides in a right triangle. For non-right triangles or other shapes like parallelograms, the sine function is used to find heights or areas. The sine of an angle \(\theta\) is calculated by the formula \(\sin(\theta)\), where \(\theta\) can be measured in either degrees or radians. In our exercise, we found \(\sin(1.745) \approx 0.9848\) using a calculator. This value was then used to compute the area of the parallelogram. Calculators and mathematical tables typically include values for \(\sin(\theta)\), making it straightforward to apply in various math problems, further demonstrating the versatile usefulness of the sine function in analytical geometry.