Problem 19
Question
$$ \begin{aligned} &\text { Let } x, y, \text { and } z \text { denote the angles of an arbitrary triangle. }\\\ &\text { Find the maximum value of } \sin x \sin y \sin z \text { . } \end{aligned} $$
Step-by-Step Solution
Verified Answer
The maximum value of \( \sin x \sin y \sin z \) in a triangle is \( \frac{3\sqrt{3}}{8} \).
1Step 1: Recognize Triangle Angle Sum
Since \( x, y, \) and \( z \) are the angles of a triangle, we know that their sum is \( 180^\circ \) or \( \pi \) radians. This gives us an equation: \( x + y + z = \pi \).
2Step 2: Apply Trigonometric Product Property
We know from trigonometric identities that the product \( \sin x \sin y \sin z \) can be tied to the product of sines of angles that sum up to a constant value (\( \pi \)). In particular, the maximum value occurs when the angles are closest to \( 60^\circ \) or \( \frac{\pi}{3} \) radians each.
3Step 3: Set Angles for Maximization
To maximize \( \sin x \sin y \sin z \), assume the triangle is equilateral with \( x = y = z = \frac{\pi}{3} \).
4Step 4: Calculate the Product of Sines
For an equilateral triangle, each angle is \( \frac{\pi}{3} \). Thus, \( \sin x = \sin y = \sin z = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \). The product is \( \left( \frac{\sqrt{3}}{2} \right)^3 = \frac{3\sqrt{3}}{8} \).
Key Concepts
Triangle Sum TheoremSine FunctionMaximum Value of Trigonometric Functions
Triangle Sum Theorem
In any triangle, the sum of the interior angles is always constant. Known as the Triangle Sum Theorem, this principle states that the interior angles of a triangle add up to \( 180^\circ \) or \( \pi \) radians. This theorem is fundamental in both Euclidean geometry and trigonometry.
This fixed sum allows precision in calculations involving angles within triangles. For instance, if you know two angles of a triangle, you can always determine the third angle by subtracting the sum of the known angles from \( 180^\circ \).
In equations, this is expressed as:
This fixed sum allows precision in calculations involving angles within triangles. For instance, if you know two angles of a triangle, you can always determine the third angle by subtracting the sum of the known angles from \( 180^\circ \).
In equations, this is expressed as:
- \( x + y + z = 180^\circ \) or \( x + y + z = \pi \) (in radians)
Sine Function
The sine function is an essential part of trigonometry, mainly concerned with right triangles and the unit circle. It relates the length of the side opposite the angle to the hypotenuse in right triangles. The sine function is periodic, with a cycle of \( 360^\circ \) or \( 2\pi \) radians. In mathematical terms, for an angle \( \theta \), the sine is defined as:
The sine of an angle fluctuates between -1 and 1. In many trigonometric problems, like the one presented, understanding the behavior of the sine function is crucial, especially when dealing with the product of sine values for angles within triangles.
- \( \sin(\theta) = \frac{\text{Opposite side}}{\text{Hypotenuse}} \)
The sine of an angle fluctuates between -1 and 1. In many trigonometric problems, like the one presented, understanding the behavior of the sine function is crucial, especially when dealing with the product of sine values for angles within triangles.
Maximum Value of Trigonometric Functions
Finding the maximum product of trigonometric functions, like sine, often requires understanding their behavior over specific intervals. The maximum value of \( \sin x \sin y \sin z \) occurs under particular conditions when dealing with angles that sum to a constant value.
For the sine function, this involves aligning the angles to optimize their sine values. Since sine reaches its peak at \( 90^\circ \), but constrained by triangle conditions – requiring sums to \( 180^\circ \) – the trick is to balance values.
In a triangle with interior angles \( x, y, \) and \( z \), maximizing \( \sin x \sin y \sin z \) happens when all three angles are as large as possible, ideally equal. This balance is achieved in an equilateral triangle, where each angle is \( 60^\circ \) or \( \frac{\pi}{3} \) radians.
If \( x = y = z = \frac{\pi}{3} \), each sine value would be \( \frac{\sqrt{3}}{2} \), maximizing their product at \( \frac{3\sqrt{3}}{8} \). Recognizing such symmetry and properties of angles is pivotal in optimizing trigonometric products.
For the sine function, this involves aligning the angles to optimize their sine values. Since sine reaches its peak at \( 90^\circ \), but constrained by triangle conditions – requiring sums to \( 180^\circ \) – the trick is to balance values.
In a triangle with interior angles \( x, y, \) and \( z \), maximizing \( \sin x \sin y \sin z \) happens when all three angles are as large as possible, ideally equal. This balance is achieved in an equilateral triangle, where each angle is \( 60^\circ \) or \( \frac{\pi}{3} \) radians.
If \( x = y = z = \frac{\pi}{3} \), each sine value would be \( \frac{\sqrt{3}}{2} \), maximizing their product at \( \frac{3\sqrt{3}}{8} \). Recognizing such symmetry and properties of angles is pivotal in optimizing trigonometric products.
Other exercises in this chapter
Problem 18
Determine whether \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} \sin x}{x^{2}+y^{2}}\) exists.
View solution Problem 18
Sketch the level curve \(f(x, y)=c\). \(f(x, y)=2 y-\cos x ; c=0,1,2\)
View solution Problem 19
Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=e^{x y}
View solution Problem 19
Compute \(d w / d t\) $$ w=\sin x y^{2} z^{3} ; x=3 t, y=t^{1 / 2}, z=t^{1 / 3} $$
View solution