What is the range of [tex]$y=-5 \sin (x)$[/tex]?
A. all real numbers
B. [tex]$-5 \leq y \leq 5$[/tex]
C. all real numbers [tex]$-\frac{5}{2} \leq y \leq \frac{5}{2}$[/tex]
D. all real numbers [tex]$-1 \leq y \leq 1$[/tex]
E. all real numbers [tex]$-\frac{1}{5} \leq y \leq \frac{1}{5}$[/tex]
Answer (1)
The range of sin ( x ) is − 1 ≤ sin ( x ) ≤ 1 .
Multiplying by -5 reverses the inequality signs: 5 ≥ − 5 sin ( x ) ≥ − 5 .
Rewriting gives − 5 ≤ − 5 sin ( x ) ≤ 5 .
Therefore, the range of y = − 5 sin ( x ) is − 5 ≤ y ≤ 5 .
Explanation
Understanding the Problem We are asked to find the range of the function y = − 5 sin ( x ) . The range represents all possible output values of the function.
Range of Sine Function We know that the sine function, sin ( x ) , oscillates between -1 and 1, inclusive. That is, − 1 ≤ sin ( x ) ≤ 1
Multiplying by -5 Now, we need to find the range of − 5 sin ( x ) . To do this, we multiply the inequality by -5. Remember that when we multiply an inequality by a negative number, we must reverse the inequality signs. So, we have − 5 ( − 1 ) ≥ − 5 sin ( x ) ≥ − 5 ( 1 ) which simplifies to 5 ≥ − 5 sin ( x ) ≥ − 5
Final Range We can rewrite this inequality as − 5 ≤ − 5 sin ( x ) ≤ 5 Since y = − 5 sin ( x ) , we can substitute y into the inequality: − 5 ≤ y ≤ 5 Thus, the range of the function y = − 5 sin ( x ) is all real numbers between -5 and 5, inclusive.
Examples
Understanding the range of trigonometric functions like y = − 5 sin ( x ) is useful in many real-world applications. For example, in electrical engineering, alternating current (AC) voltage and current can be modeled using sinusoidal functions. If the voltage is given by V ( t ) = − 5 sin ( t ) , then knowing the range − 5 ≤ V ( t ) ≤ 5 tells us the maximum and minimum voltage values, which is crucial for designing circuits and ensuring components can handle the voltage levels. Similarly, in physics, simple harmonic motion, such as the motion of a pendulum or a mass on a spring, can be described using sinusoidal functions, and the range helps determine the maximum displacement from the equilibrium position.
