Problem 23 - Algebra 2 and Trigonometry

Question

Let \(A B C D\) be a parallelogram with \(A B=c, B C=a,\) and \(\mathrm{m} \angle B=\theta .\) a. Write a formula for the area of parallelogram \(A B C D\) in terms of \(c, a,\) and \(\theta\) . b. For what value of \(\theta\) does parallelogram \(A B C D\) have the greatest area?

Step-by-Step Solution

Answer

a. The area is \(ac \sin \theta\). b. The greatest area is when \(\theta = 90^\circ\).

Step 1: Understanding Parallelogram Area Formula

The formula for the area of a parallelogram can be expressed as the product of the base and the height. For parallelogram \(ABCD\), with \(AB = c\) and \(BC = a\), we use the formula for area: \( \, \text{Area} = c \times (a \sin \theta) \, \), where \(a \sin \theta\) is the height corresponding to base \(c\) since height is perpendicular to the base.

Step 2: Deriving the Area Formula

Substituting the values into the formula, we get \(\text{Area} = c \times a \sin \theta = ac \sin \theta \). This formula expresses the area of the parallelogram in terms of \(c, a,\) and \(\theta\).

Step 3: Maximizing the Area

To find the \(\theta\) that gives the greatest area, we need the maximum value of \(\sin \theta\), since \(\text{Area} = ac \sin \theta\). The maximum value of \(\sin \theta\) is 1, which occurs when \(\theta = 90^\circ\).

Step 4: Verifying the Maximum Condition

When \(\theta = 90^\circ\), \(\sin \theta = 1\) and the area of the parallelogram becomes \(ac\cdot 1 = ac\), which is indeed the greatest possible area for this configuration given the sides \(a\) and \(c\).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. Understanding trigonometry is crucial when studying areas of shapes like parallelograms, where angles play a vital role. In the exercise, the angle \( \theta \) is central to determining the height and thus the area of the parallelogram.
Trigonometry uses functions like sine, cosine, and tangent to relate angles to side lengths:

• The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse.
• The cosine function gives the ratio of the adjacent side to the hypotenuse.
• Trigonometric identities often simplify the task of finding unknown angles or side lengths.

With a parallelogram, thinking about one angle \( \theta \) helps determine the vertical height from the base to the opposite side, crucial for calculating area.

Sine Function

The sine function is a key player in trigonometry, particularly when working out areas in geometric shapes. It's denoted as \( \sin(\theta) \) and gives the vertical component of an angle, which is essential when calculating heights in non-right-angled shapes.
Here are some important points about the sine function:

• Values of \( \sin(\theta) \) range between -1 and 1.
• The maximum value of \( \sin(\theta) \) is 1, achieved at \( \theta = 90^\circ \) (or \( \pi/2 \) radians).
• The function is periodic, with a period of \(2\pi\), meaning it repeats its values in a regular cycle.

In the context of the parallelogram \(ABCD\), \( \sin(\theta) \) is used to determine the effective height perpendicular to the base \(c\), making it integral to the area calculation \( ac \sin(\theta) \). Understanding that \( \theta \) affects the sine value helps clarify why certain angles yield larger area values.

Maximizing Area

When it comes to finding the maximum area of a shape like a parallelogram, optimizing the variables involved is key. For parallelogram \(ABCD\), the formula \( ac \sin(\theta) \) shows that both \(a\), \(c\), and \(\sin(\theta)\) directly influence the area.
Here's how to approach maximizing the parallelogram's area:

• Focus on \( \sin(\theta) \). Since its maximum possible value is 1, the largest area occurs when \( \theta = 90^\circ \).
• At this angle, the height is simply the line segment length \( a \), perfectly vertical to the base \( c \).
• Thus, \( ac \sin(90^\circ) = ac \) represents the maximum area, since \( \sin(90^\circ) = 1 \).

By adjusting \( \theta \) to this optimal angle, the parallelogram essentially becomes a rectangle, which inherently maximizes the interior space using given side lengths \(a\) and \(c\). Recognizing how these components interact to influence area can be pivotal in solving problems efficiently.