When dealing with mathematical expressions, it is common to encounter questions about their maximum or minimum values. In this case, we are asked to determine the maximum value of the expression 3-7cos(5x). Let’s delve into this question and explore the concept in detail.<\/p>\n
The expression 3-7cos(5x) consists of two parts: the constant term 3 and the trigonometric function cos(5x). The maximum value of cos(5x) is 1, which occurs when the angle 5x is equal to 0 degrees or 360 degrees. To find the maximum value of the entire expression, we substitute the maximum value of the cosine function into it.<\/p>\n
What is the maximum value of 3-7cos(5x)?<\/b><\/p>\n
<\/p>\n
The maximum value of 3-7cos(5x) is 3-7(1) = 3-7 = -4.<\/p>\n
Therefore, the maximum value of the expression 3-7cos(5x) is -4.<\/p>\n
1. What is cosine?<\/b><\/p>\n
<\/p>\n
Cosine is a trigonometric function that represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle.<\/p>\n
2. How does cosine vary with different angles?<\/b><\/p>\n
<\/p>\n
Cosine varies between -1 and 1, oscillating periodically as the angle changes.<\/p>\n
3. How does the cosine function behave?<\/b><\/p>\n
<\/p>\n
The cosine function oscillates between its maximum value of 1 and minimum value of -1 over a complete revolution.<\/p>\n
4. What is the period of the cosine function?<\/b><\/p>\n
<\/p>\n
The period of the cosine function is 360 degrees or 2\u03c0 radians.<\/p>\n
5. What is the maximum value of cosine?<\/b><\/p>\n
<\/p>\n
The maximum value of the cosine function is 1, which occurs when the angle is 0 degrees or 360 degrees.<\/p>\n
6. What is the minimum value of cosine?<\/b><\/p>\n
<\/p>\n
The minimum value of the cosine function is -1, which occurs when the angle is 180 degrees.<\/p>\n
7. How does the negative sign affect the cosine function?<\/b><\/p>\n
<\/p>\n
The negative sign in front of the cosine function reflects it vertically, causing it to be reflected about the x-axis.<\/p>\n
8. What happens when we multiply the cosine function by a constant?<\/b><\/p>\n
<\/p>\n
Multiplying the cosine function by a constant results in stretching or shrinking the graph vertically, thus affecting the amplitude.<\/p>\n
9. How does the value of x affect the maximum value of 3-7cos(5x)?<\/b><\/p>\n
<\/p>\n
The value of x affects the angle 5x, which ultimately determines the maximum value of the expression. A change in x will displace the maximum point, but the maximum value itself will remain -4.<\/p>\n
10. Are there any constraints on the value of x for the maximum value to occur?<\/b><\/p>\n
<\/p>\n
No, there are no constraints on the value of x for the maximum value of -4 to occur; it is valid for all real values of x.<\/p>\n
11. Can the maximum value be positive?<\/b><\/p>\n
<\/p>\n
No, the maximum value of 3-7cos(5x) is -4, and it cannot be positive.<\/p>\n
12. What is the significance of the maximum value in this context?<\/b><\/p>\n
<\/p>\n
The maximum value of the expression represents the highest point that it can reach. Understanding this value helps in analyzing the behavior and range of the function.<\/p>\n
In conclusion, the maximum value of 3-7cos(5x) is -4. Understanding the concept of maximum values in trigonometric functions is fundamental in analyzing mathematical expressions and their behavior.<\/p>\n","protected":false},"excerpt":{"rendered":"
When dealing with mathematical expressions, it is common to encounter questions about their maximum or minimum values. In this case, we are asked to determine the maximum value of the expression 3-7cos(5x). Let’s delve into this question and explore the concept in detail. The expression 3-7cos(5x) consists of two parts: the constant term 3 and … <\/p>\n