Maximum of a $\sin$ and $\cos$ combo

Calculus Level pending

Numerically find the maximum of

$(12 + 4 \cos \alpha + 7 \cos \beta )^2 + (-5 + 3 \sin \alpha + 4 \sin \beta)^2$

Inspiration

The answer is 560.1.

1 solution

Piotr Idzik
Dec 11, 2020

I used simulated annealing and the function provided in the python module scipy.optimize.

# -*- coding: utf-8 -*-
"""
solution of:
    https://brilliant.org/problems/maximum-of-a-sin-and-cos-combo/
"""

import scipy.optimize
import numpy


def get_val(in_vec):
    """
    returns the value of the function from:
        https://brilliant.org/problems/maximum-of-a-sin-and-cos-combo/
    """
    alpha = in_vec[0]
    beta = in_vec[1]
    val_a = 12+4*numpy.cos(alpha)+7*numpy.cos(beta)
    val_b = -5+3*numpy.sin(alpha)+4*numpy.sin(beta)
    return val_a**2+val_b**2


LOWER = [0]*2
UPPER = [2*numpy.pi]*2
RES = scipy.optimize.dual_annealing(
    lambda x: -get_val(x),
    bounds=list(zip(LOWER, UPPER)),
    maxiter=10000)
print(-RES.fun)

Maximum of a sin ⁡ \sin sin and cos ⁡ \cos cos combo

The answer is 560.1.

1 solution

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Maximum of a $\sin$ and $\cos$ combo