Using the graph of y=cos(x), list all values of x on the interval [-11π/3,11π/6] that satisfy the ordered pair of (x,-1/radical2).

- List all the values of (x,-1/radical2)

(Type an exact answer using π as needed. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the​ expression.)

- Please, ​someone, ​help me. Show work step by step so that I can learn it. I have no idea where to start, so please be specific on the steps on how to solve it. Thank You

1/√2 is the cosine of 45º (π/4)

... that is the reference angle

-1/√2 is the cosine of
... ±3π/4
... add or subtract multiples of 8π/4 to find the values on the interval

x values...-13π/4, -11π/4, -5π/4, -3π/4, 3π/4, 5π/4

To find the values of x on the interval [-11π/3, 11π/6] that satisfy the ordered pair (x, -1/√2), we need to find the x-values for which cos(x) equals -1/√2.

Step 1: Convert -1/√2 to its decimal approximation.
-1/√2 ≈ -0.7071

Step 2: Find the angles whose cosine is approximately -0.7071 using the inverse cosine function (cos^(-1)).
cos^(-1)(-0.7071) ≈ 3π/4 + 2πn and 5π/4 + 2πn, where n is an integer.

Step 3: Setting up the inequalities using the interval [-11π/3, 11π/6].
-11π/3 ≤ x ≤ 11π/6

Step 4: Substitute the solutions from step 2 into the inequalities to find the valid values of x within the given interval.
-11π/3 ≤ 3π/4 + 2πn ≤ 11π/6
-11π/3 - 3π/4 ≤ 2πn ≤ 11π/6 - 3π/4
-44π/12 ≤ 2πn ≤ 66π/12
-22π/6 ≤ πn ≤ 11π/6

-11π/6 ≤ 5π/4 + 2πn ≤ 11π/6
-11π/6 - 5π/4 ≤ 2πn ≤ 11π/6 - 5π/4
-44π/12 ≤ 2πn ≤ 22π/12
-22π/6 ≤ πn ≤ 11π/6

Step 5: Simplify the inequalities to find the valid values of n.
-22π/6 ≤ πn ≤ 11π/6

Solving this inequality will give us the valid values of n within the given interval.

Step 6: Solve the inequality -22π/6 ≤ πn ≤ 11π/6 for n.
Divide all terms by π: -11/3 ≤ n ≤ 11/6

Step 7: Since n is an integer, the valid values for n are -4, -3, -2, -1, 0, and 1.

Step 8: Substitute the values of n back into the equations from step 2 to find the corresponding x-values.
For n = -4:
x = 3π/4 + 2π(-4)
= 3π/4 - 8π
= π/4

For n = -3:
x = 3π/4 + 2π(-3)
= 3π/4 - 6π
= -3π/4

For n = -2:
x = 3π/4 + 2π(-2)
= 3π/4 - 4π
= -π/4

For n = -1:
x = 3π/4 + 2π(-1)
= 3π/4 - 2π
= π/4

For n = 0 (x-value we already found earlier):
x = 3π/4 + 2π(0)
= 3π/4
≈ 2.3562

For n = 1 (x-value we already found earlier):
x = 3π/4 + 2π(1)
= 3π/4 + 2π
= 11π/4

Step 9: List the values of x in the ordered pairs (x, -1/√2):
(π/4, -1/√2)
(-3π/4, -1/√2)
(-π/4, -1/√2)
(π/4, -1/√2)
(2.3562, -1/√2)
(11π/4, -1/√2)

Therefore, the values of x on the interval [-11π/3, 11π/6] that satisfy the ordered pair (x, -1/√2) are:
π/4, -3π/4, -π/4, 2.3562, 11π/4.

To find the values of x on the interval [-11π/3,11π/6] that satisfy the ordered pair (x, -1/√2) on the graph of y = cos(x), we need to find the x-values for which cos(x) = -1/√2.

Here is the step-by-step process to solve the problem:

Step 1: Rewrite -1/√2 in terms of the cosine function.
We know that cos(π/4) = 1/√2. Since cosine is an even function, cos(x) = -1/√2 at x = -π/4.

Step 2: Determine the period of the cosine function.
The graph of y = cos(x) repeats its pattern every 2π units. So to find all the values of x on the interval [-11π/3, 11π/6], we can find the solution in the range of one cycle (0 to 2π) and then add or subtract multiples of 2π.

Step 3: Solve cos(x) = -1/√2.
On the interval [0,2π], cos(x) = -1/√2 at x = 7π/4, which satisfies the ordered pair (x, -1/√2).

Step 4: Add or subtract multiples of 2π.
Since the cosine function has a period of 2π, we can add or subtract multiples of 2π to find other solutions. In this case, -11π/3 is less than 0, so let's subtract 2π until we reach a value in the range [0,2π].

-11π/3 - 2π = -17π/3
-17π/3 - 2π = -23π/3
-23π/3 - 2π = -29π/3

Step 5: Check if the values are within the given interval.
Since -29π/3 is less than the lower bound of the interval (-11π/3), we stop because we cannot subtract another 2π.

Step 6: Organize the solutions.
The solutions we found are x = -π/4 and x = -17π/3.

Therefore, the values of x on the interval [-11π/3,11π/6] that satisfy the ordered pair (x, -1/√2) are x = -π/4 and x = -17π/3.