$\sqrt{\frac{\cos 2 x}{1+\sin ^2 x}}$
Answer (1)
1 + 2 t a n 2 x 1 − t a n 2 x for − 4 π + nπ ≤ x ≤ 4 π + nπ .
Explanation
Problem Analysis We are given the expression 1 + s i n 2 x c o s 2 x and we want to analyze it.
Rewriting cos 2x First, we rewrite cos 2 x using the identity cos 2 x = cos 2 x − sin 2 x . This gives us 1 + s i n 2 x c o s 2 x − s i n 2 x .
Rewriting 1 Next, we rewrite 1 as sin 2 x + cos 2 x . Substituting this into the denominator, we get s i n 2 x + c o s 2 x + s i n 2 x c o s 2 x − s i n 2 x = c o s 2 x + 2 s i n 2 x c o s 2 x − s i n 2 x .
Dividing by cos^2 x Now, we divide both the numerator and the denominator by cos 2 x . This gives us 1 + 2 t a n 2 x 1 − t a n 2 x .
Analyzing the Domain To determine the domain of the expression, we need to ensure that the expression inside the square root is non-negative, i.e., 1 + s i n 2 x c o s 2 x ≥ 0 . Since 1 + sin 2 x is always positive, we only need to consider when cos 2 x ≥ 0 .
Solving for x The inequality cos 2 x ≥ 0 holds when − 2 π + 2 nπ ≤ 2 x ≤ 2 π + 2 nπ , where n is an integer. Dividing by 2, we get − 4 π + nπ ≤ x ≤ 4 π + nπ .
Final Expression and Domain Therefore, the expression 1 + s i n 2 x c o s 2 x is defined for − 4 π + nπ ≤ x ≤ 4 π + nπ , and can be rewritten as 1 + 2 t a n 2 x 1 − t a n 2 x .
Examples
Understanding trigonometric expressions and their domains is crucial in fields like physics and engineering. For instance, when analyzing the motion of a pendulum, the angle it makes with the vertical can be modeled using trigonometric functions. Determining the valid range of these angles ensures that the model accurately represents the physical constraints of the pendulum's movement, preventing it from exceeding its maximum possible displacement. This ensures the stability and predictability of the system being analyzed.
