Question

In a triangle \(\mathrm{ABC}\), maximum value of \(\sin \mathrm{A}+\sin \mathrm{B}+\sin \mathrm{C}\) and \(\sin \mathrm{A} \sin \mathrm{B} \sin \mathrm{C}\) are respectively equal to (a) \(\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{8}\) (b) \(\frac{3 \sqrt{3}}{2}, \frac{3 \sqrt{3}}{8}\) (c) \(\frac{3 \sqrt{3}}{4}, \frac{3 \sqrt{3}}{8}\) (d) \(\frac{\sqrt{3}}{4}, \frac{\sqrt{3}}{2}\)

Short answer

Answer: The maximum values of sinA + sinB + sinC and sinA * sinB * sinC for a given triangle ABC are \(\frac{3\sqrt{3}}{2}\) and \(\frac{3\sqrt{3}}{8}\), respectively.

Step-by-step solution

Write the sum of angles of the triangle

In a triangle, the sum of interior angles is always 180 degrees, so we can write: A + B + C = 180°

Express sinC in terms of sinA and sinB

From the equation above, we have C = 180° - A - B, so we can now express sinC as: sinC = sin(180° - A - B) Using the sine addition formula, we have: sinC = sin(180° - A)cosB - cos(180° - A)sinB sinC = sinAcosB - cosAsinB Now we can rewrite the expression sinA + sinB + sinC as: sinA + sinB + sinAcosB - cosAsinB

Find the maximum value of sinA + sinB + sinC

To maximize the expression sinA + sinB + sinAcosB - cosAsinB, we will use the fact that for any angle x, -1 ≤ sinx ≤ 1 and -1 ≤ cosx ≤ 1. Notice that sinA and sinB are at their maximum when A and B are 60° each. Hence, the maximum value of sinA + sinB + sinC is: sin60° + sin60° + sin60° = \frac{3\sqrt{3}}{2} So, the correct option is (b) for the maximum value of sinA + sinB + sinC.

Find the maximum value of sinA * sinB * sinC

We will now find the maximum value of sinA * sinB * sinC using the expression for sinC we found in Step 2: sinA * sinB * sinC = sinA * sinB * (sinAcosB - cosAsinB) To maximize this expression, we will once again use the fact that A and B are 60° each. Hence, the maximum value of sinA * sinB * sinC is: sin60° * sin60° * sin60° = \frac{\sqrt{3}}{2} * \frac{\sqrt{3}}{2} * \frac{1}{2} = \frac{3\sqrt{3}}{8} So, the correct option is (b) for the maximum value of sinA * sinB * sinC. Therefore, the answer is (b) \(\frac{3\sqrt{3}}{2}, \frac{3\sqrt{3}}{8}\).

Key concepts

Triangle Properties

Understanding the properties of a triangle is essential when working with trigonometric identities and optimizing angles. One of the fundamental properties is that the sum of the internal angles in any triangle is always 180 degrees. This rule is crucial in calculating and manipulating the trigonometric functions of the angles within the triangle.

• Given angles A, B, and C in triangle ABC, they must satisfy the equation: \[ A + B + C = 180° \]
• This property helps us express one angle in terms of the other two, which is particularly useful when we try to solve equations involving trigonometric identities.
• Understanding triangle properties allows us to substitute one angle when needed, simplifying complex trigonometric expressions.

This foundational property ensures that as long as two angles are known, the third can be easily computed, providing a gateway to further explorations in trigonometric functions.

Sine Function

The sine function is a crucial component in trigonometry, representing the ratio of the length of the side opposite an angle to the hypotenuse in a right triangle. In general, the sine of an angle helps us to understand the relationship between the angles and sides in any triangle.

• The sine function is periodic with a period of 360°, and its values range from -1 to 1.
• In a triangle, you can often find \[ \sin C = \sin (180° - A - B) \] which simplifies due to the sine angle subtraction identity: \[ \sin (180° - x) = \sin x \]
• In equilateral triangles or special cases such as angles of 60°, the sine function reaches a maximum under certain conditions. For example, \[ \sin 60° = \frac{\sqrt{3}}{2} \]

The property of the sine function being bounded allows us to predict the behavior of angles in a triangle and solve optimizations involving trigonometric identities, as seen in joint expressions such as \(\sin A + \sin B + \sin C\).

Angle Optimization

Angle optimization in trigonometry involves finding the angle measures that yield the maximum or minimum values for trigonometric expressions. This is particularly important when evaluating expressions like \( \sin A + \sin B + \sin C \) or the product \( \sin A * \sin B * \sin C \).

• By the property that trigonometric functions are bounded, specifically sine and cosine, these expressions are maximized when angles A, B, and C result in optimal sine values.
• For instance, with each sine term having a maximum value of \( \sin 60° = \frac{\sqrt{3}}{2} \), assuming A = B = 60°, and thus C = 60°, collectively maximizes both the sum and product of these sine functions.
• In optimization problems within a triangle, angle values are adjusted within the confines set by triangle properties to reach those optimal trigonometric values.

This approach is useful in various applications, including maximizing structural efficiencies or solving problems where peak trigonometric outputs are required, ultimately showcasing the harmony between geometric properties and trigonometric behavior.