How to solve for unknown theta angle

Question

So I am trying to write a script that solves for an unknown angle. The equation will be of the form of:

P*sin(theta)+Q*cos(theta)=G

Where P, Q, and G are known values and theta needs to be found. I was wondering if there was a way to do this with numpy or math modules. I am really new to python so trying to figure out what modules are out there and how to use them.

From googling around I heard about fsolve but unsure how to use it. Is there a library for solving for unknowns?

• Mike 'Pomax' Kamermans: if P, Q, and G are known, just solve it directly? You have one function, and one unknown: rewrite your function to an expression of theta? And if you don't know, there's always wolframalpha and friends.
• lastchance: Try theta=math.asin(G/math.sqrt(P**2+Q**2))-math.atan2(Q,P)
• Mike 'Pomax' Kamermans: @lastchance inline code is single backticks, not triple backticks (or double backticks if the code itself has single backticks in it). Also, if you're going to give them an actual formula instead of letting them do the math, write an answer and explain why you're giving them that formula.
• lastchance: Thank-you, @Mike 'Pomax' Kamermans. I still manage to fluff some of my responses here. I've written out a cursory answer.
• dan04: Recall the trig identity P * sin(θ) + Q * cos(θ) = √(P² + Q²) * sin(θ + atan2(Q, P)).

Answer 1

OK, as requested. The aim is to write the LHS as A.sin(theta+phi). Then we would be able to solve from

sin(theta+phi)=G/A

or

theta=asin(G/A)-phi ( exists provided abs(G) <= abs(A) )

So you need to find A and phi such that

P.sin(theta)+Q.cos(theta) = A.sin(theta+phi)
 = A[ sin(theta)cos(phi)+cos(theta)sin(phi) ]

Equate coefficients of cos(theta) and sin(theta):

A.sin(phi)=Q, A.cos(phi)=P

Square and add:

A²=P²+Q²

Divide:

tan(phi)=Q/P ( phi=atan2(Q,P) gives correct quadrant )

In code, with a check:

import math
def get_theta( P, Q, G ):
 ''' Returns (one value of) theta such that P.sin(theta) + Q.cos(theta) = G'''
 return math.asin( G / math.sqrt( P ** 2 + Q ** 2 ) ) - math.atan2( Q, P )
P, Q, G = 4, 3, 2
theta = get_theta( P, Q, G )
print( "theta = ", theta, " radians, or ", theta * 180 / math.pi, "degrees" )
print( "Check: LHS = ", P * math.sin( theta ) + Q * math.cos( theta ) )

Output:

theta = -0.2319842627257963 radians, or -13.291719167642189 degrees
Check: LHS = 2.0000000000000004

Important notes:

(1) It is your responsibility to check there IS a solution ( sqrt(P²+Q²)>=abs(G) )

(2) If there is one solution then there are an infinity of others; n.pi+(-1)ⁿasin(G/A)-phi

• fender: Awesome thanks for answering I will give this a try!

Answer 2

Have you tried to use Scipy?

import numpy as np
from scipy.optimize import fsolve
#Define P, Q, G
def equation(theta):
 return P * np.sin(theta) + Q * np.cos(theta) - G
result = fsolve(equation, 0)
print(result)

• fender: This is what I was looking for, thank you. I will read up on the documentation for Scipy. Why is it that we have to specify that we only want to import fsolve from scipy? is it for optimization/speed benefits?