Home › Trigonometry

How to maximize a product of four sine terms

Pair symmetric angles

The identity $sin (x + a) sin (x - a) = (cos (2 a) - cos (2 x)) /2$ is useful when factors come in plus/minus pairs. Once the product is a square, its maximum and minimum follow from the range of cosine.

Problem

Let $f (x) = sin (x + π /4) sin (x + 3 π /4) sin (x - π /4) sin (x - 3 π /4)$ . Find the largest value of (1) $f (x)$ and (2) $f (x) - f (y)$ , where $x$ and $y$ vary independently.

Answer

Both maxima are

1/4

The function ranges from 0 to 1/4, and x and y can be chosen independently.

Step-by-step solution

1. Pair the opposite shifts

For $a = π /4$ , $cos (2 a) = 0$ , so $sin (x + π /4) sin (x - π /4) = - cos (2 x) /2$ . The same identity with $a = 3 π /4$ gives another factor of $- cos (2 x) /2$ .

2. Bound the product

Therefore $f (x) = cos^{2} (2 x) /4$ . As a square of a cosine ranges from 0 to 1, the largest value of $f$ is $1/4$ and the smallest is 0.

3. Maximize the difference

Choose $x$ where $f (x) = 1/4$ and $y$ where $f (y) = 0$ . Then $f (x) - f (y) = 1/4$ , and no larger difference is possible.

Add to Chrome - solve your own

Screenshot any problem on your screen and get the steps. Chrome on a computer; free.