Here's how to solve this problem:

1. Solve cos(θ) = sin(θ):

   • This equation is true when θ = 45° + 180° * n, where n is any integer. This is because sine and cosine are equal at 45 degrees, and then repeat this relationship every 180 degrees due to their periodicity and phase difference.

2. Incorporate the Given Information

   • We are given that θ is 83 times greater than 90 degrees, meaning θ > 83 * 90°. θ > 7470°.

3. Find the Appropriate Angle:

   • We need to find an integer n such that θ = 45° + 180° * n and θ > 7470°.

   • Let's solve for n : 45 + 180*n* > 7470

   • 180*n* > 7425

   • n > 41.25

   • The smallest integer n that satisfies this condition is n = 42.

   • Therefore, θ = 45° + 180° * 42 = 45° + 7560° = 7605°.

4. Calculate 60 * θ

   • 60 * θ = 60 * 7605° = 456300°