Question

Suppose x is a real number with 0 ≤ x ≤ π. Set a = sin(x - π), b = cos(x + π), c = sin(2 - x), and d = cos(2 + x). Arrange the values a, b, c, and d in increasing order (least to greatest), and explain how you determined their order.

Answer 1

To arrange the values a, b, c, and d in increasing order, substitute the given values of x into the expressions and calculate their numerical values: a = sin(x - π), b = cos(x + π), c = sin(2 - x), d = cos(2 + x). The values arranged in increasing order are b, d, a, c: -1 ≤ -0.4161 ≤ 0 ≤ 0.9093

Step-by-step explanation:

To arrange the values a, b, c, and d in increasing order, we need to calculate their numerical values using the given expressions:

a = sin(x - π)

b = cos(x + π)

c = sin(2 - x)

d = cos(2 + x)

Since 0 ≤ x ≤ π, we can substitute x = π into the expressions for a and b:

a = sin(π - π) = 0

b = cos(π + π) = -1

To calculate the values of c and d, we need to substitute x = 0 into the expressions:

c = sin(2 - 0) = sin(2) ≈ 0.9093

d = cos(2 + 0) = cos(2) ≈ -0.4161

Therefore, the values arranged in increasing order are b, d, a, c, which means: -1 ≤ -0.4161 ≤ 0 ≤ 0.9093.