For the function $y=-3+4 \cos \left(\frac{5 \pi}{6}(x+4)\right)$, what is the minimum value?
Answer (1)
The problem asks for the minimum value of the function y = − 3 + 4 cos ( 6 5 π ( x + 4 ) ) .
The minimum value of the cosine function is -1.
Substitute -1 for the cosine term: y min = − 3 + 4 ( − 1 ) .
Calculate the minimum value: − 7 .
Explanation
Analyze the problem We are given the function y = − 3 + 4 cos ( 6 5 π ( x + 4 ) ) and we want to find its minimum value.
Minimum value of cosine The cosine function, cos ( θ ) , always has values between -1 and 1, i.e., − 1 ≤ cos ( θ ) ≤ 1 . Therefore, the minimum value of cos ( θ ) is -1.
Calculate the minimum value To find the minimum value of the given function, we need to find the minimum value of the cosine term. Since the minimum value of cos ( 6 5 π ( x + 4 ) ) is -1, we substitute -1 into the function: y min = − 3 + 4 ( − 1 ) y min = − 3 − 4 y min = − 7
Final Answer Therefore, the minimum value of the function y = − 3 + 4 cos ( 6 5 π ( x + 4 ) ) is -7.
Examples
Understanding the minimum and maximum values of trigonometric functions is crucial in many fields. For example, in electrical engineering, the voltage in an AC circuit can be modeled using a cosine function. Knowing the minimum voltage ensures that devices connected to the circuit are designed to handle the lowest possible voltage level, preventing malfunctions or damage. Similarly, in acoustics, the intensity of a sound wave can be modeled using trigonometric functions, and determining the minimum intensity helps in designing noise cancellation systems or ensuring audibility in different environments.
