To solve the trigonometric equation \( \sin(x) - 2024 \cos(x) = 0 \):

1. **Rearrange the equation**:
   \[
   \sin(x) = 2024 \cos(x)
   \]

2. **Divide both sides by \( \cos(x) \) (assuming \( \cos(x) \neq 0 \))**:
   \[
   \tan(x) = 2024
   \]

3. **Solve for \( x \)**:
   \[
   x = \tan^{-1}(2024) + n\pi, \quad n \in \mathbb{Z}
   \]
   Here, \( \tan^{-1}(2024) \) is the principal value of the arctangent of 2024, and \( n\pi \) represents the general solution for the tangent function because the tangent function is periodic with period \( \pi \).

4. **Approximate the principal value**:
   \[
   x \approx \tan^{-1}(2024)
   \]

   You can calculate the value of \( \tan^{-1}(2024) \) using a calculator or by software.

### Key Points
- The general solution is \( x = \tan^{-1}(2024) + n\pi \), where \( n \) is any integer.
- This solution includes all possible values of \( x \) that satisfy the equation.

Would you like to know the detailed calculation of \( \tan^{-1}(2024) \), or do you have any questions?

Here are 5 related questions you might find interesting:
1. What are the properties of the tangent function and its inverse?
2. How do we derive the general solution for trigonometric equations?
3. What is the significance of the period of the tangent function?
4. How does the periodic nature of trigonometric functions affect their solutions?
5. How can we solve similar trigonometric equations with different coefficients?

**Tip:** When solving trigonometric equations, remember to consider the periodic nature of the trigonometric functions, which leads to multiple solutions.

Learn how to solve the trigonometric equation sin(x) - 2024 cos(x) = 0 step-by-step. Understand the use of trigonometric identities and periodicity of trigonometric functions to find all solutions. Suitable for advanced high school students and anyone interested in trigonometric equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation