Problem 18 - Calculus with Concepts in Calculus

Question

Let \(x\) and \(y\) denote the acute angles of a right triangle. Find the maximum value of \(\sin x \sin y\).

Step-by-Step Solution

Answer

The maximum value of \(\sin x \sin y\) is \(\frac{1}{2}\).

Step 1: Understand the Relationship Between Angles

Since \(x\) and \(y\) are the acute angles of a right triangle, they add up to \(90^\circ\). Therefore, we have the relationship: \(x + y = 90^\circ\). This implies \(y = 90^\circ - x\).

Step 2: Express the Product in Terms of One Angle

Using the relationship \(y = 90^\circ - x\), we express \(\sin y\) as \(\sin(90^\circ - x) = \cos x\). Hence, the expression \(\sin x \sin y\) can be rewritten as \(\sin x \cos x\).

Step 3: Use a Trigonometric Identity

Recall the trigonometric identity: \(\sin(2x) = 2\sin x\cos x\). This allows us to express \(\sin x \cos x\) as \(\frac{1}{2} \sin(2x)\). Therefore, \(\sin x \sin y = \frac{1}{2} \sin(2x)\).

Step 4: Determine Maximum Value of the Sine Function

The function \(\sin(2x)\) has a maximum value of 1, which occurs when \(2x = 90^\circ\) or \(2x = 270^\circ\), but given that \(x\) and \(y\) are acute angles, we only consider \(2x = 90^\circ\). This implies \(x = 45^\circ\) and \(y = 45^\circ\).

Step 5: Calculate Maximum Value

Substituting \(x = 45^\circ\) into the equation \(\frac{1}{2} \sin(2x)\), we get \(\frac{1}{2} \sin(90^\circ) = \frac{1}{2} \times 1 = \frac{1}{2}\).

Step 6: Conclusion: Verify the Solution

Since both angles \(x\) and \(y\) are equal to \(45^\circ\), which satisfies the initial condition of being acute angles adding up to 90 degrees, the solution is verified as correct.

Key Concepts

Maximum Value

When we talk about finding the maximum value of a trigonometric function, we are looking for the largest possible output that the function can achieve. In the context of the problem, we are given the expression \(\sin x \sin y\), where \(x\) and \(y\) are acute angles of a right triangle.

• The product \(\sin x \sin y\) needs to be maximized.
• Using the identity \(\sin x \cos x = \frac{1}{2} \sin(2x)\), we can simplify our expression to \(\frac{1}{2} \sin(2x)\).
• The sine function, \(\sin(\theta)\), reaches its maximum value of 1 when \(\theta = 90^\circ\).

For this problem, since \(x\) and \(y\) are acute angles, the angle for maximum value is \(2x = 90^\circ\),which means \(x = 45^\circ\) and correspondingly, \(y = 45^\circ\). The maximum of \(\sin x \sin y\) is then \(\frac{1}{2} \times 1 = \frac{1}{2}\).

Right Triangle

A right triangle is a type of triangle where one of the angles is exactly \(90^\circ\). This makes the other two angles acute, meaning they must be less than \(90^\circ\). In this case, our angles are \(x\) and \(y\).

• The sum of the angles in a triangle always adds up to \(180^\circ\).
• Given the right angle, \(x + y = 90^\circ\).
• This property is crucial because it places a restriction on how large or small \(x\) and \(y\) can be independently.

A right triangle not only helps us define these angles but is also fundamental in deriving relationships and solving problems that involve trigonometry. The relationship between \(x\) and \(y\) as complements is noted by \(y = 90^\circ - x\), providing an essential step in evaluating expressions such as \(\sin x \sin y\).

Acute Angles

Acute angles are angles that are less than \(90^\circ\). In any right triangle, the acute angles will always sum to \(90^\circ\) because the third angle is \(90^\circ\) itself.

• In our exercise, both \(x\) and \(y\) are acute angles.
• Since they must sum to \(90^\circ\), knowing one of the angles immediately determines the other.
• This means that if \(x = 45^\circ\), then naturally \(y = 45^\circ\) as well.

Understanding the nature and properties of acute angles is vital for solving problems like this. Recognizing that \(x\) and \(y\) are complementary allows us to use trigonometric identities such as \(\sin(90^\circ - x) = \cos x\). These identities simplify expressions and help in locating maximum and minimum function values where acute angles are involved.