To solve the equation \( 2 \tan(x) + 3 = 1 \), follow these steps:

### Step 1: Isolate \(\tan(x)\)
Subtract 3 from both sides:
\[
2 \tan(x) = 1 - 3
\]
\[
2 \tan(x) = -2
\]

### Step 2: Solve for \(\tan(x)\)
Now, divide both sides by 2:
\[
\tan(x) = \frac{-2}{2} = -1
\]

### Step 3: Find the general solution for \(\tan(x) = -1\)
The tangent function equals \(-1\) at specific angles. The principal solution for \(\tan(x) = -1\) is:
\[
x = \frac{3\pi}{4} + n\pi, \quad \text{where} \ n \in \mathbb{Z}
\]
This represents the general solution, where \(n\) is any integer, accounting for the periodicity of the tangent function.

Would you like more details on this, or do you have any further questions?

---

**Related Questions:**
1. How does the periodicity of the tangent function affect its solutions?
2. What are the properties of the tangent function at different quadrants?
3. How would the solution change if the equation were \(2\tan(x) + 3 = 5\)?
4. Can you solve \(2\sin(x) + 3 = 1\) for \(x\) similarly?
5. What is the graphical interpretation of solving \(\tan(x) = -1\)?

**Tip:** Remember that tangent has a period of \(\pi\), meaning solutions repeat every \(\pi\) radians.

Learn how to solve the trigonometric equation 2 tan(x) + 3 = 1 with a detailed, step-by-step solution. This problem covers key trigonometric concepts such as the tangent function, its periodicity, and solving equations. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation