Question

If x − 6 < δ, then cos2(x) − 3/4 < 0.05.

a) δ > cos^(-1)(1/2) + 6
b) δ < cos^(-1)(1/2) - 6
c) δ > cos^(-1)(1/2) - 6
d) δ < cos^(-1)(1/2) + 6

Answer 1

To solve the inequality cos^2(x) - (3/4) < 0.05, we can start by factoring out the common factor of cos^2(x) - (3/4) to get (cos(x) - 1/2)(cos(x) + 3/2) < 0.05. The solution is δ > cos^(-1)(1/2) + 6.

Step-by-step explanation:

To solve the inequality cos^2(x) - (3/4) < 0.05, we can start by factoring out the common factor of cos^2(x) - (3/4) to get (cos(x) - 1/2)(cos(x) + 3/2) < 0.05. Now, we have to consider two cases:

Case 1: cos(x) - 1/2 < 0 and cos(x) + 3/2 > 0

Solve the first inequality by adding 1/2 to both sides: cos(x) < 1/2. The solution to this inequality is cos^(-1)(1/2) + 6. Therefore, δ > cos^(-1)(1/2) + 6.

Case 2: cos(x) - 1/2 > 0 and cos(x) + 3/2 < 0

Solve the second inequality by subtracting 3/2 from both sides: cos(x) < -3/2. However, there are no solutions to this inequality because the cosine function's range is between -1 and 1. Therefore, we only need to consider Case 1.