https://file.aiquickdraw.com/mathbot/file/1724926677517nj2il9mw.jpg

The equation provided is:

\[
\tan(3x + 50^\circ) = \tan(7x - 25^\circ)
\]

To solve this equation in degrees, we again use the property of the tangent function, which states that if \(\tan A = \tan B\), then:

\[
A = B + n \cdot 180^\circ
\]

where \(n\) is any integer.

### Case 1: \(A = B\)
\[
3x + 50^\circ = 7x - 25^\circ
\]

Solving for \(x\):

\[
50^\circ + 25^\circ = 7x - 3x
\]
\[
75^\circ = 4x
\]
\[
x = \frac{75^\circ}{4} = 18.75^\circ
\]

### Case 2: \(A = B + 180^\circ\)
\[
3x + 50^\circ = 7x - 25^\circ + 180^\circ
\]

Solving for \(x\):

\[
50^\circ + 25^\circ = 7x - 3x + 180^\circ
\]
\[
75^\circ = 4x + 180^\circ
\]
\[
-105^\circ = 4x
\]
\[
x = \frac{-105^\circ}{4} = -26.25^\circ
\]

### Case 3: \(A = B + 360^\circ\)
This case involves adding another 180° to the previous case:

\[
3x + 50^\circ = 7x - 25^\circ + 360^\circ
\]
\[
50^\circ + 25^\circ = 7x - 3x + 360^\circ
\]
\[
75^\circ = 4x + 360^\circ
\]
\[
-285^\circ = 4x
\]
\[
x = \frac{-285^\circ}{4} = -71.25^\circ
\]

### Summary of Solutions:
The solutions in degrees are:

\[
x = 18.75^\circ, \quad x = -26.25^\circ, \quad x = -71.25^\circ
\]

Would you like to explore more cases or have any questions about this solution?

Here are five related questions for further practice:

1. What is the general solution for \( \tan(A) = \tan(B) \) in degrees?
2. How do you account for all possible solutions in trigonometric equations?
3. Can you solve for \(x\) if the equation was \( \tan(4x + 20^\circ) = \tan(2x - 10^\circ) \)?
4. What is the impact of adding 180° to the angle in tangent equations?
5. How would the solution change if the equation involved sine or cosine instead of tangent?

**Tip:** Always consider adding multiples of the period of the trigonometric function (like 180° for tangent) to find additional solutions.

Learn how to solve the trigonometric equation \( \tan(3x + 50^\circ) = \tan(7x - 25^\circ) \) step-by-step with detailed solutions. Understand the properties of tangent functions and apply them to find all solutions in degrees. Suitable for high school students and those 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