91Ó°ÊÓ

Trigonometry involves various formulas that make solving problems easier, and one such set of formulas are the double angle formulas. These formulas provide relationships for the sine, cosine, and tangent of double angles. This is particularly useful in problems where one needs to simplify the expressions involving trigonometric functions of multiplied angles.

A common double angle formula for sine, for example, is: \[\sin(2\theta) = 2\sin(\theta)\cos(\theta)\]. This identity essentially states that the sine of twice any angle can be expressed as the double of the product of the sine and cosine of the original angle.

In our exercise, we used this formula to transform the inequality \(|\sin (2 \theta)| \leq 2|\sin \theta|\) into a more manageable form. This manipulation is vital as it leverages the bounds of the trigonometric functions (both sine and cosine range from -1 to 1) to prove the inequality without complex calculations.

Understanding and applying double angle formulas are not only important for solving trigonometric inequalities but also in simplifying integrals, solving differential equations, and analyzing wave functions in physics.

The absolute value of a real number is its distance from zero on the number line, regardless of direction. It is always non-negative, as distance cannot be negative. The notation for the absolute value of a number 'x' is written as \(|x|\).

Some important properties of absolute value that are relevant to our exercise include:

• \(|-x| = |x|\), reflecting the fact that distance is always positive.
• If \(0 \leq x \leq y\), then \(|x| \leq |y|\), depicting that larger numbers have greater or equal absolute value.
• \(|xy| = |x||y|\), which tells us that the absolute value of a product is the product of the absolute values.

In the context of the given problem, we know that both \(\sin(\theta)\) and \(\cos(\theta)\) produce outputs within the range of -1 to 1, and considering the property mentioned last, the absolute value of their product will also lie within this interval. This intrinsic property of absolute values is crucial to validate the inequality \(|\sin(\theta)\cos(\theta)| \leq |\sin(\theta)|\) we derived in the solution.

Trigonometric functions are the cornerstone of trigonometry and are used to relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine, cosine, and tangent.

Each of these functions has specific properties and ranges:

• The sine function \(\sin(\theta)\) and cosine function \(\cos(\theta)\) both have a range from -1 to 1.
• The tangent function is the ratio of sine to cosine and can take any real value from \(-\infty\) to \(\infty\).

In our inequality problem, we are dealing with the sine and cosine functions. The fact that their values are constrained between -1 and 1, inclusive, is crucial since it allows us to ascertain that the absolute value of their product \(|\sin(\theta)\cos(\theta)|\) is limited to the same range. This property simplifies our understanding of the inequality and aids in proving it.

A strong grasp of the behavior of trigonometric functions not only supports solving inequalities but is also fundamental in analyzing periodic phenomena, wave mechanics, and in the field of engineering for analyzing AC circuits.