Do the equations
(a) |sin x|=sin x+3, (b) |tan x|=tan x+3 have any roots?

Answer

To determine whether the equations \( | \sin x | = \sin x + 3 \) and \( | \tan x | = \tan x + 3 \) have any roots, we will analyze each equation step by step.

### Part (a): \( | \sin x | = \sin x + 3 \)

1. **Understanding the absolute value**: The absolute value function \( | \sin x | \) can be defined in two cases:
- Case 1: \( \sin x \geq 0 \)
- Case 2: \( \sin x < 0 \)

2. **Case 1: \( \sin x \geq 0 \)**
In this case, \( | \sin x | = \sin x \). Thus, the equation becomes:
\[ \sin x = \sin x + 3 \]
Simplifying this gives:
\[ 0 = 3 \]
This is a contradiction. Therefore, there are no solutions in this case.

3. **Case 2: \( \sin x < 0 \)**
Here, \( | \sin x | = -\sin x \). The equation becomes:
\[ -\sin x = \sin x + 3 \]
Rearranging gives:
\[ -2\sin x = 3 \quad \Rightarrow \quad \sin x = -\frac{3}{2} \]
The sine function has a range of \([-1, 1]\), and \(-\frac{3}{2}\) is outside this range. Thus, there are no solutions in this case either.

### Conclusion for Part (a):
The equation \( | \sin x | = \sin x + 3 \) has **no roots**.

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### Part (b): \( | \tan x | = \tan x + 3 \)

1. **Understanding the absolute value**: Similar to the previous case, we analyze \( | \tan x | \) in two cases:
- Case 1: \( \tan x \geq 0 \)
- Case 2: \( \tan x < 0 \)

2. **Case 1: \( \tan x \geq 0 \)**
In this case, \( | \tan x | = \tan x \). The equation becomes:
\[ \tan x = \tan x + 3 \]
Simplifying gives:
\[ 0 = 3 \]
This is again a contradiction. Therefore, there are no solutions in this case.

3. **Case 2: \( \tan x < 0 \)**
Here, \( | \tan x | = -\tan x \). The equation becomes:
\[ -\tan x = \tan x + 3 \]
Rearranging gives:
\[ -2\tan x = 3 \quad \Rightarrow \quad \tan x = -\frac{3}{2} \]
The tangent function has a range of \((- \infty, \infty)\), so \(-\frac{3}{2}\) is within this range. Thus, there are solutions in this case.

### Conclusion for Part (b):
The equation \( | \tan x | = \tan x + 3 \) has **roots** since \( \tan x = -\frac{3}{2} \) is a valid solution.

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### Summary of Results:
- Part (a): **No roots**
- Part (b): **Has roots** (specifically, \( \tan x = -\frac{3}{2} \))