Trigonometric Functions

To find the minimum value of the expression 5cos⁡(2)+5sin⁡(2)5cos(2x)+5sin(2x), we can use trigonometric identities to simplify it.
First, let's express the given terms in a single trigonometric function using the Pythagorean identity (sin⁡2()+cos⁡2()=1sin2(x)+cos2(x)=1):
5cos⁡(2)+5sin⁡(2)=5(cos⁡(2)+sin⁡(2))5cos(2x)+5sin(2x)=5(cos(2x)+sin(2x))
Now, we'll use a trigonometric identity sin(x+4π​)=2​1​(cos(x)+sin(x))) to combine the terms inside the parentheses:
5(cos(2x)+sin(2x))=52​sin(2x+4π​)
Since the sine function has values ranging from -1 to 1, the minimum value of ​sin(2x+4π​) is -5252​.
Therefore, the minimum value of the given expression5cos(2x)+5sin(2x) is −52−52​.
To find the minimum value of the expression 5^cos(2x) + 5^sin(2x), we can use the fact that the range of both the cosine and sine functions is between -1 and 1.
Since 5^cos(2x) and 5^sin(2x) are both positive for any value of x, the minimum value of the expression occurs when both terms equal their minimum value of 1.
Therefore, the minimum value of 5^cos(2x) + 5^sin(2x) is 1 + 1 = 2.