EXT $x+x+\operatorname{seg} x+\frac{1}{4} x$ the follow Graf Loon modifies where $322+2 x^2+5 \leq x^2$
Answer (1)
Simplify the inequality: x 2 ≤ − 327 , which gives x = ± i 327 .
Simplify the expression: x + x + sec x + 4 1 x = 4 9 x + sec x .
Substitute x = i 327 : 4 9 ( i 327 ) + sec ( i 327 ) .
Substitute x = − i 327 : 4 9 ( − i 327 ) + sec ( − i 327 ) = − 4 9 i 327 + sec ( i 327 ) .
The final answers are 4 9 i 327 + sec ( i 327 ) and − 4 9 i 327 + sec ( i 327 )
Explanation
Understanding the Problem We are asked to simplify the expression x + x + seg x + 4 1 x given the condition 322 + 2 x 2 + 5 ≤ x 2 . The expression 'seg x' is not a standard mathematical notation. We will assume 'seg x' means 'sec x' (secant of x).
Simplifying the Inequality First, let's simplify the inequality 322 + 2 x 2 + 5 ≤ x 2 :
322 + 2 x 2 + 5 ≤ x 2 327 + 2 x 2 ≤ x 2 327 ≤ x 2 − 2 x 2 327 ≤ − x 2 x 2 ≤ − 327
Solving for x Now, let's solve for x :
x 2 ≤ − 327 x = ± − 327 x = ± i 327
Simplifying the Expression Next, we simplify the expression x + x + sec x + 4 1 x :
x + x + sec x + 4 1 x = 2 x + 4 1 x + sec x = 4 8 x + 4 1 x + sec x = 4 9 x + sec x
Substituting the values of x Now, we substitute x = i 327 and x = − i 327 into the simplified expression 4 9 x + sec x .
For x = i 327 :
4 9 ( i 327 ) + sec ( i 327 )
For x = − i 327 :
4 9 ( − i 327 ) + sec ( − i 327 ) Since sec ( − z ) = sec ( z ) , the second expression is − 4 9 ( i 327 ) + sec ( i 327 )
Final Answer Thus, the two possible values of the expression are 4 9 i 327 + sec ( i 327 ) and − 4 9 i 327 + sec ( i 327 ) .
Examples
Complex numbers and trigonometric functions are used in electrical engineering to analyze AC circuits. The impedance of a circuit element can be represented as a complex number, and the voltage and current can be expressed as sinusoidal functions with complex exponentials. The secant function is used in various signal processing applications, such as designing filters and analyzing the frequency response of systems. Understanding these concepts is crucial for designing and analyzing electrical circuits and signal processing systems.
