Find the greatest possible value of

5cos(x)+6sin(x).

I know that it would be a good idea to look for identities that are in the form of acos(x)+bsin(x), but I can't seem to find any. Other than that I don't really know where to go with this problem. Any help would be much appreciated!

Yes, there are lots of such formulas. One is at

http://www.uwfox.uwc.edu/users/tnyman/TRIGConcepts/Asinx+Bcosx.pdf

where you will see that the amplitude of your function is √(5^2+6^2) = √61

Ah, the classic "greatest possible value" question. You're absolutely right about looking for identities in the form of acos(x) + bsin(x).

Let's combine our powers of observation and humor, shall we? Remember the good old square root of (a^2 + b^2)? That's the Pythagorean theorem. Let's reinterpret 5cos(x) + 6sin(x) as the sides of a right triangle.

We have the 5cos(x) as one side and 6sin(x) as another. To find the hypotenuse, let's use the Pythagorean theorem: √[(5cos(x))^2 + (6sin(x))^2].

Now, let me let you in on a secret. We want to maximize the hypotenuse, right? Well, when does the Pythagorean theorem result in the largest possible value for the hypotenuse?

Drumroll, please... when the triangle is a right triangle with a 90-degree angle! So, we need to find the angle x that gives us a right triangle.

Now, when does cos(x) = 0 (vertical side equals 0), and sin(x) = 1 (horizontal side equals 1)? Bingo! It's when x = π/2 (or 90 degrees).

Plugging this value into the cosine and sine terms, we get 5cos(π/2) + 6sin(π/2) = 5(0) + 6(1) = 6.

So, the greatest possible value of 5cos(x) + 6sin(x) is 6. Ta-da! Hope that helps lighten up your math journey a bit.

To find the greatest possible value of the expression 5cos(x) + 6sin(x), we can use the concept of the maximum value of a trigonometric function.

Let's rewrite the expression as A*cos(x) + B*sin(x), where A = 5 and B = 6.

We know that the maximum value of this expression (A*cos(x) + B*sin(x)) is given by √(A^2 + B^2), according to the maximum value formula for this general form of expression.

Therefore, the maximum value of 5cos(x) + 6sin(x) is √(5^2 + 6^2) = √(25 + 36) = √61.

Hence, the greatest possible value of the expression 5cos(x) + 6sin(x) is √61.

To find the greatest possible value of 5cos(x) + 6sin(x), we can rewrite it using the formula for combining trigonometric functions known as the sine and cosine addition formulas.

Let's start by expressing 5cos(x) + 6sin(x) in terms of sine and cosine. By the cosine addition formula:

5cos(x) + 6sin(x) = √(5^2 + 6^2) * [ (5/√(5^2 + 6^2)) * cos(x) + (6/√(5^2 + 6^2)) * sin(x) ]

Simplifying the expression further:

5cos(x) + 6sin(x) = √61 * [ (5/√61) * cos(x) + (6/√61) * sin(x) ]

Now, we can recognize that the expression inside the brackets is in the form of acos(x) + bsin(x). So, we can rewrite it as a single trigonometric function using a new angle φ:

(5/√61) * cos(x) + (6/√61) * sin(x) = √61 * sin(φ)

Now, we can rewrite the entire expression as:

5cos(x) + 6sin(x) = √61 * √61 * sin(φ)
= 61 * sin(φ)

To find the greatest possible value of 5cos(x) + 6sin(x), we need to find the maximum value of sin(φ).

Since the sine function attains its maximum value of 1 at 90 degrees (or π/2 radians), we can conclude that the greatest possible value of 5cos(x) + 6sin(x) is:

61 * sin(90°) = 61

Therefore, the greatest possible value of 5cos(x) + 6sin(x) is 61.