# Solutions of a System of Two Trigonometric Equations

Introduction: A system of two trigonometric equations involves two equations with at least one common variable. Solving such systems requires finding values for the variables that satisfy both equations simultaneously. Here are key concepts and steps for solving a system of two trigonometric equations:

### 1. System of Two Trigonometric Equations:

• Example:
• Consider the system $\mathrm{sin}\left(x\right)=\frac{1}{2}$ and $\mathrm{cos}\left(x\right)=\frac{\sqrt{3}}{2}$

### 2. Solving Strategies:

• Use Common Variable:

• Identify a common variable in both equations. In the example, $x$ is common to both $\mathrm{sin}\left(x\right)$ and $\mathrm{cos}\left(x\right)$.
• Apply Trigonometric Identities:

• If applicable, use trigonometric identities to simplify or relate the two equations. For example, $\mathrm{cos}\left(x\right)=\sqrt{1-{\mathrm{sin}}^{2}\left(x\right)}$ is a useful identity.
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• Use Algebraic Techniques:

• Employ algebraic techniques such as substitution or elimination to combine or manipulate the equations. Substitute one equation into the other to reduce the system to a single equation.

### 3. Example: Solving a System of Trigonometric Equations:

• Given System:

• $\mathrm{sin}\left(x\right)=\frac{1}{2}$ and $\mathrm{cos}\left(x\right)=\frac{\sqrt{3}}{2}$.
• Solution Steps:

1. Use Common Variable:

• The common variable is $x$, which appears in both equations.
2. Apply Trigonometric Identities:

• Consider the identity $\mathrm{cos}\left(x\right)=\sqrt{1-{\mathrm{sin}}^{2}\left(x\right)}$ to relate the two equations.
1.
2. Substitute or Eliminate:

• Substitute the expression for $\mathrm{cos}\left(x\right)$ in terms of $\mathrm{sin}\left(x\right)$ into the first equation: $\mathrm{sin}\left(x\right)=\frac{1}{2}$
• Solve for $x$, obtaining $x=\frac{\pi }{6}+2n\pi$ and $x=\frac{5\pi }{6}+2n\pi$.
3. Verify Solutions:

• Check that the solutions satisfy both original equations.

### 4. General Solution and Periodicity:

• General Solution:

• Express the general solution by incorporating an integer parameter. For example, $x=\frac{\pi }{6}+2n\pi$ represents all possible solutions.
• Periodicity Consideration:

• Be aware of the periodicity of trigonometric functions. Solutions may repeat at intervals of $2\pi$ or $\pi$.