Problem 13 - Thomas Calculus in SI Units

Question

Two sides of a triangle have lengths \(a\) and \(b\), and the angle between them is \(\theta\). What value of \(\theta\) will maximize the triangle's area? (Hint: \(A=(1 / 2) a b \sin \theta\).)

Step-by-Step Solution

Answer

The triangle's area is maximized when \(\theta = 90^\circ\).

Step 1: Understand the Area Formula

The area of a triangle given two sides and the included angle is computed using the formula \(A = \frac{1}{2}ab\sin\theta\). Here, \(a\) and \(b\) are the lengths of the sides, and \(\theta\) is the angle between them.

Step 2: Maximize the Sine Function

The area \(A\) is directly proportional to \(\sin\theta\). To maximize the area, we need to maximize \(\sin\theta\). The maximum value of the sine function is 1.

Step 3: Determine the Optimal Angle

The sine of an angle reaches its maximum value of 1 when the angle \(\theta\) is \(90^\circ\) or \(\frac{\pi}{2}\) radians. Therefore, to maximize the area, \(\theta\) should be \(90^\circ\).

Key Concepts

Triangle Area

The area of a triangle can be calculated using various methods, depending on the information available about the triangle. When you know the lengths of two sides and the measure of the angle between them, you can use the trigonometric formula for the area. The formula is given by:\[A = \frac{1}{2}ab\sin\theta,\]where \(a\) and \(b\) are the side lengths, and \(\theta\) is the included angle between these sides.
This method is particularly useful for non-right triangles, allowing you to bypass the need to know the height directly. This area formula comes from the general concept that the area of a triangle is \(\frac{1}{2}\) times its base times its height. In this case, \(\sin\theta\) effectively represents the height when the line from one vertex is perpendicular to the opposite side.

• The formula is applicable only when the angle is included between the sides \(a\) and \(b\).
• A triangle with a known angle and two sides makes it perfect for this trigonometric method.

Sine Function

The sine function is a fundamental concept in trigonometry and is crucial for solving problems involving triangles. It relates the angle in a right-angled triangle to the ratio of the length of the opposite side to the hypotenuse. The sine function is periodic and oscillates between -1 and 1, making it perfect for various wave phenomena.
In the context of maximizing the area of a triangle, understanding the sine function is key. The sine of an angle \(\theta\) varies between 0 and 1 for angles between 0° and 90° (or 0 and \(\frac{\pi}{2}\) radians).

• For angles beyond 90°, \(\sin\theta\) begins to decrease, thus providing the largest value at 90°.
• Because \(\sin\theta = 1\) gives the largest possible effect on the triangle area formula, it directly maximizes the area of a triangle formed with sides \(a\) and \(b\).

Recognizing that the sine function is optimal at 90° helps in determining the configuration where the triangle reaches its fullest potential in terms of area.

Optimal Angle

To find the optimal angle that maximizes the area of the triangle, we focus on the properties of the sine function. Since the triangle's area is a function of \(\sin\theta\), we aim for the highest value of this sine function within the valid range of triangle angles.
For any triangle, the maximum value \(\sin\theta\) can achieve is 1, and this occurs when \(\theta = 90^\circ\) or \(\frac{\pi}{2}\) radians.

• Thus, the optimal angle for maximizing the triangle's area is a right angle, where the triangle takes the form of a right triangle.
• By setting \(\theta\) to 90°, you guarantee that the sides \(a\) and \(b\) are perpendicular, leading to the maximum possible area.

This principle is not only significant in pure geometry but also in practical applications where maximizing space or area is a priority. The optimal point showcases how the sine function and the angle work together for maximizing geometrical properties.