$\sqrt{\frac{\cos 2 x}{1+\sin ^2 x}}$
Answer (1)
Rewrite cos 2 x as cos 2 x − sin 2 x and 1 as sin 2 x + cos 2 x .
Simplify the expression to 1 + 2 t a n 2 x 1 − t a n 2 x .
Determine the domain by solving the inequality 1 + 2 t a n 2 x 1 − t a n 2 x ≥ 0 , which simplifies to − 1 ≤ tan x ≤ 1 .
The solution to the inequality is − 4 π + kπ ≤ x ≤ 4 π + kπ , where k is an integer. Therefore, the expression is defined for x in the interval [ − 4 π + kπ , 4 π + kπ ] .
Explanation
Understanding the Expression We are given the expression 1 + s i n 2 x c o s 2 x and we need to analyze it.
Rewriting cos 2x We can rewrite cos 2 x using the identity cos 2 x = cos 2 x − sin 2 x . So the expression becomes 1 + s i n 2 x c o s 2 x − s i n 2 x .
Replacing 1 Now, we can rewrite 1 as sin 2 x + cos 2 x . The expression then becomes 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 Next, 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 Now, let's analyze the domain of the expression. For the square root to be real, the expression inside the square root must be non-negative, i.e., 1 + 2 t a n 2 x 1 − t a n 2 x ≥ 0 . Since 1 + 2 tan 2 x is always positive, we need 1 − tan 2 x ≥ 0 , which means tan 2 x ≤ 1 , or − 1 ≤ tan x ≤ 1 .
Determining the Interval This implies − 4 π + kπ ≤ x ≤ 4 π + kπ , where k is an integer. This means x must lie in the interval [ − 4 π , 4 π ] or any interval obtained by adding integer multiples of π to these bounds.
Examples
Understanding trigonometric expressions and their domains is crucial in fields like physics and engineering. For instance, when analyzing the motion of a pendulum or the behavior of alternating current in an electrical circuit, you often encounter expressions involving trigonometric functions. Determining the valid range of angles or time intervals ensures that the mathematical model accurately represents the physical system, preventing nonsensical results like imaginary lengths or infinite currents. This ensures the stability and predictability of designs and analyses.
