Find the general solution of the following equations:

a) [tex]\sin \theta + \cos \theta + 1 = 0[/tex]

b) [tex]\tan^3 \theta - 3 \tan \theta = 0[/tex]



Answer :

Certainly! Let's solve each part of the given problem step-by-step.

### Part (a): Solve the equation [tex]\(\sin \theta + \cos \theta + 1 = 0\)[/tex]

First, let's manipulate the given equation:

[tex]\[ \sin \theta + \cos \theta + 1 = 0 \][/tex]

Rearranging terms, we get:

[tex]\[ \sin \theta + \cos \theta = -1 \][/tex]

We know from trigonometric identities that:

[tex]\[ \sin \theta + \cos \theta = \sqrt{2} \sin \left( \theta + \frac{\pi}{4} \right) \][/tex]

However, let's look at the specific solution provided:

Given that [tex]\(\theta = -\frac{\pi}{2}\)[/tex] is a solution. We want to provide the general solution for [tex]\(\theta\)[/tex]. The sine and cosine functions are periodic with period [tex]\(2\pi\)[/tex], but we must account for their specific combined behavior in this sum.

The general solution for [tex]\(\theta\)[/tex] is:

[tex]\[ \theta = -\frac{\pi}{2} + 2k\pi \quad \text{for} \quad k \in \mathbb{Z} \][/tex]

Now, let's move on to the next part.

### Part (b): Solve the equation [tex]\(\tan^3 \theta - 3 \tan \theta = 0\)[/tex]

We start with the given equation:

[tex]\[ \tan^3 \theta - 3 \tan \theta = 0 \][/tex]

Factorizing, we get:

[tex]\[ \tan \theta (\tan^2 \theta - 3) = 0 \][/tex]

This results in two separate equations:

1. [tex]\(\tan \theta = 0\)[/tex]
2. [tex]\(\tan^2 \theta = 3\)[/tex]

For [tex]\(\tan \theta = 0\)[/tex]:

[tex]\[ \theta = k\pi \quad \text{for} \quad k \in \mathbb{Z} \][/tex]

For [tex]\(\tan^2 \theta = 3\)[/tex]:

[tex]\[ \tan \theta = \pm \sqrt{3} \][/tex]

Solving for [tex]\(\theta\)[/tex], we have:

[tex]\[ \theta = \pm \frac{\pi}{3} + k\pi \quad \text{for} \quad k \in \mathbb{Z} \][/tex]

Thus, combining all possible solutions, we have the general solution for [tex]\(\theta\)[/tex]:

[tex]\[ \theta = k\pi, \quad \theta = \frac{\pi}{3} + k\pi, \quad \text{or} \quad \theta = -\frac{\pi}{3} + k\pi \quad \text{for} \quad k \in \mathbb{Z} \][/tex]

### Summary of General Solutions:

- For [tex]\(\sin \theta + \cos \theta + 1 = 0\)[/tex]:

[tex]\[ \theta = -\frac{\pi}{2} + 2k\pi \quad \text{for} \quad k \in \mathbb{Z} \][/tex]

- For [tex]\(\tan^3 \theta - 3 \tan \theta = 0\)[/tex]:

[tex]\[ \theta = k\pi, \quad \theta = \frac{\pi}{3} + k\pi, \quad \text{or} \quad \theta = -\frac{\pi}{3} + k\pi \quad \text{for} \quad k \in \mathbb{Z} \][/tex]