Solving equations of the form ##a\sin\theta+b\cos\theta = c##

AI Thread Summary
The discussion focuses on solving the equation sin(θ) + 2cos(θ) = 1 for the angle θ. The initial approach involves transforming the equation into a form that allows the use of trigonometric identities, leading to a general solution expressed in terms of n. However, the solution diverges from the book's answer, which suggests a different form involving a positive acute angle γ. An alternate method confirms that the solutions align closely, but the original method's correctness is questioned due to discrepancies in sign and the necessity of checking all potential solutions. Ultimately, the discussion emphasizes the importance of verifying each solution and understanding the underlying trigonometric relationships.
brotherbobby
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Homework Statement
Give the general solution to the equation : ##\sin\theta+2\cos\theta=1##
Relevant Equations
1. If ##\sin\theta = \sin\alpha## where ##\alpha## is the minimum positive angle whose "##\sin##" is the same as the "##\sin##" of the angle ##\theta##, then ##\boldsymbol{\theta=n\pi+(-1)^n \alpha}##
2. If ##\cos\theta = \cos\alpha## where ##\alpha## is the minimum positive angle whose "##\cos##" is the same as the "##\cos##" of the angle ##\theta##, then ##\boldsymbol{\theta=2n\pi\pm \alpha}##
3. ##\sin^2\theta = 1-\cos^2\theta##
Problem statement : Given the equation ##\sin\theta+2\cos\theta=1##, find the general solution for the angle ##\theta##.

Attempt : For the general case where we have ##a\sin\theta+b\cos\theta=c##, the line of approach is to take ##a=r\cos\alpha## and ##b=r\sin\alpha## wherein we will have ##r=(a^2+b^2)^{1/2}## and ##\alpha= \tan^{-1}\frac{b}{a}##.

For the equation ##\sin\theta+2\cos\theta=1##, let ##1=r\cos\alpha## and ##2=r\sin\alpha##. These yield ##r=\sqrt{5}## and ##\alpha = \tan^{-1}2= 63.43^{\circ}##.
The given equation reduces to ##r\sin(\theta+\alpha)=1\Rightarrow \underline{\sin(\theta+\alpha)=\frac{1}{\sqrt{5}}} = \sin\beta\;\text{(say)}##, which gives ##\beta = \sin^{-1}(1/\sqrt{5}) = 26.57^{\circ}##.
The underlined equation can be solved using Relevant Equation(1) above to give : ##\theta+\alpha = n\pi+(-1)^n\beta\Rightarrow \boxed{\theta = n\pi -\tan^{-1}2+(-1)^n \sin^{-1}(1/\sqrt{5})}##.

Check :
(a) On putting ##n=0##, we have ##\theta_0 = -\tan^{-1}2+\sin^{-1}(1/\sqrt{5}) = -36.9^{\circ}## which satisfies the given equation ##\large{\checkmark}##.
(b) On putting, ##n=1##, we have ##\theta_1 = \pi -\tan^{-1}2-\sin^{-1}(1/\sqrt{5}) = 90^{\circ}## which satisfies the given equation ##\large{\checkmark}##.
(c) On putting, ##n=2##, we have ##\theta_2 = 2\pi -\tan^{-1}2+\sin^{-1}(1/\sqrt{5}) = 323.13^{\circ}## which satisfies the given equation ##\large{\checkmark}##.
[I assume that my general solution above (in box) is correct and will hold for all values of ##n \in \mathbb{Z}##.]

The issue : The answer in the book is given as ##\boxed{2n\pi+\pi/2\; \text{OR}\; 2n\pi-\gamma}\; \text{where}\; \gamma\; \text{is a positive acute angle whose sine is}\; 3/5##.

Check (Book's answer) :
(a) Putting ##n=0##, we have ##\theta_0 = 90^{\circ}\;\text{OR}\; \theta_0 = -\gamma = -36.9^{\circ}##. (These match my answers above though for different values of ##n##)##\large{\checkmark}##.
(b) Putting ##n=1##, we have ##\theta_1 = 450^{\circ}\;\text{OR}\; \theta_1 =2\pi -\gamma = 323.1^{\circ}##. (The second one matches my answers above though for a different value of ##n##)##\large{\checkmark}##. (The first one may match too for a different value of ##n##).

Is my solution correct? I understand that unless I check for several more values of ##n## it is difficult for someone to say.

Alternate solution :
My first solution and its correctness is put further into doubt when I realize that the problem can be solved in a different way.

Given ##\sin\theta+2\cos\theta=1\Rightarrow \sin^2{\theta}= (1-2\cos\theta)^2\Rightarrow 1-\cos^2\theta=1-4\cos\theta+4\cos^2\theta\Rightarrow 4\cos\theta=5\cos^2\theta##
##\Rightarrow \cos\theta(4-5\cos\theta)=0\Rightarrow \cos\theta = 0=\cos(\pi/2)\;\text{OR} \cos\theta = 4/5=\cos\gamma##
##\Rightarrow \boxed{\theta = 2n\pi+\pi/2}\; \text{OR}\; \boxed{\theta = 2n\pi\pm \cos^{-1}(4/5)}##. This answer is almost like the one in the book with the ##\gamma##'s being the same angles. Of course I have a ##\pm## whereas the book only has a ##-##sign. I suppose the angles will match since ##\cos({-\beta})=\cos{\beta}##.

Question remains - How does the first solution look so different? Is the method alright? A help or suggestion would be welcome.
 
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You know that \cos \theta = \frac45. That means that \sin \theta = 1 - 2 \cos \theta = -\frac35. Now \sin(2n\pi + \alpha) > 0 for 0 < \alpha < \pi, so that's not going to be a solution.

Also: consider the right-angled triangle ABC where AB = 2, BC = 1 and AC = \sqrt{1 + 2^2} = \sqrt{5}. The angle at A has a sine of 1/\sqrt{5} and the angle at C has a tangent of 2. These angles must sum to \pi/2. Thus <br /> \tan^{-1}(2) + \sin^{-1} \left(\frac{1}{\sqrt{5}}\right) = \frac{\pi}{2}. By considering this triangle, you can also show that <br /> \sin\left(\tan^{-1}(2) - \sin^{-1} \left(\frac{1}{\sqrt{5}}\right)\right) = \frac 35 so the two methods do in fact agree.
 
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brotherbobby said:
Given ##\sin\theta+2\cos\theta=1\Rightarrow \sin^2{\theta}= (1-2\cos\theta)^2\Rightarrow 1-\cos^2\theta=1-4\cos\theta+4\cos^2\theta\Rightarrow 4\cos\theta=5\cos^2\theta##
##\Rightarrow \cos\theta(4-5\cos\theta)=0\Rightarrow \cos\theta = 0=\cos(\pi/2)\;\text{OR} \cos\theta = 4/5=\cos\gamma##
##\Rightarrow \boxed{\theta = 2n\pi+\pi/2}\; \text{OR}\; \boxed{\theta = 2n\pi\pm \cos^{-1}(4/5)}##. This answer is almost like the one in the book with the ##\gamma##'s being the same angles. Of course I have a ##\pm## whereas the book only has a ##-##sign. I suppose the angles will match since ##\cos({-\beta})=\cos{\beta}##.
You seemed to have skipped a few steps near the end. Applying your relevant equation #2 to ##\cos\theta = 0## gives ##\theta = 2n \pi \pm \pi/2##. Then you need to rule out the solution with the minus sign, which you may have done and just didn't mention it. But I suspected you didn't because for ##\cos\theta = 4/5=\cos\gamma##, you included the ##\pm## when solving for ##\theta##, but you didn't check if both options work. It turns out the plus sign doesn't as @pasmith noted.
 
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