Solution of Simultaneous Trigonometric Equations in One Variable

Simultaneous trigonometric equations are a set of two or more equations that involve trigonometric functions and have a common variable. The goal is to find the value(s) of this variable that satisfy all the equations at the same time. When these equations are in one variable, they can often be solved using algebraic methods, trigonometric identities, or graphical interpretations.

Understanding Trigonometric Equations

Before diving into simultaneous equations, let's understand what a trigonometric equation is. A trigonometric equation is an equation that involves trigonometric functions like sine (sin), cosine (cos), tangent (tan), etc. The solutions to these equations are the angles (usually in radians or degrees) that satisfy the equation.

Solving Simultaneous Trigonometric Equations

To solve simultaneous trigonometric equations, we often use the following methods:

1. Substitution: If one equation can be solved for one function and substituted into the other, this can simplify the system to a single equation.

2. Elimination: If adding or subtracting the equations eliminates one of the trigonometric functions, this can also reduce the system to a single equation.

3. Graphical Method: Plotting the equations on a graph can visually show the points of intersection, which correspond to the solutions.

4. Trigonometric Identities: Using identities like the Pythagorean identities, sum-to-product, or product-to-sum formulas can help simplify the equations.

5. Algebraic Manipulation: Sometimes, it's possible to manipulate the equations algebraically to find a solution.

Important Points and Differences

Aspect	Description
Number of Solutions	Trigonometric equations can have a finite number of solutions, an infinite number of solutions, or no solution at all.
Periodicity	Trigonometric functions are periodic, so solutions may repeat after a certain interval.
Domain Restrictions	Solutions must be within the domain of the trigonometric functions involved.
Range Restrictions	The range of trigonometric functions may limit the possible values of the variable.

Formulas and Identities

Some common trigonometric identities that are useful in solving simultaneous equations include:

• Pythagorean Identities:

  • $\sin^2(x) + \cos^2(x) = 1$
  • $1 + \tan^2(x) = \sec^2(x)$
  • $1 + \cot^2(x) = \csc^2(x)$

• Sum and Difference Formulas:

  • $\sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b)$
  • $\cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b)$

• Double Angle Formulas:

  • $\sin(2x) = 2\sin(x)\cos(x)$
  • $\cos(2x) = \cos^2(x) - \sin^2(x) = 2\cos^2(x) - 1 = 1 - 2\sin^2(x)$

Examples

Example 1: Substitution Method

Solve the following simultaneous equations:

1. $\sin(x) = \frac{1}{2}$
2. $\cos(x) = \frac{\sqrt{3}}{2}$

Solution:

From the first equation, we know that $x$ could be $30^\circ$ or $150^\circ$ (or in radians, $\frac{\pi}{6}$ or $\frac{5\pi}{6}$). However, we need to check which of these solutions also satisfies the second equation.

For $x = 30^\circ$ (or $\frac{\pi}{6}$), $\cos(x) = \frac{\sqrt{3}}{2}$, which matches the second equation. Therefore, $x = 30^\circ$ is a solution.

For $x = 150^\circ$ (or $\frac{5\pi}{6}$), $\cos(x) = -\frac{\sqrt{3}}{2}$, which does not satisfy the second equation. Therefore, $x = 150^\circ$ is not a solution.

Hence, the solution to the simultaneous equations is $x = 30^\circ$.

Example 2: Elimination Method

Solve the following simultaneous equations:

1. $\sin(x) + \cos(x) = 1$
2. $\sin(x) - \cos(x) = \frac{1}{2}$

Solution:

Add the two equations to eliminate $\cos(x)$:

$(\sin(x) + \cos(x)) + (\sin(x) - \cos(x)) = 1 + \frac{1}{2}$

$2\sin(x) = \frac{3}{2}$

$\sin(x) = \frac{3}{4}$

This equation has no real solution since the range of $\sin(x)$ is $[-1, 1]$ and $\frac{3}{4}$ is within this range. However, we need to find the specific angle(s) that satisfy this condition. We can use the inverse sine function to find the principal value, and then consider the periodic nature of the sine function to find all possible solutions within the desired interval.

In this case, we would find that $x$ is approximately $48.59^\circ$ or $131.41^\circ$ (or in radians, $x \approx 0.848$ or $x \approx 2.294$).

Conclusion

Solving simultaneous trigonometric equations in one variable requires a combination of algebraic manipulation, trigonometric identities, and sometimes graphical methods. Understanding the properties of trigonometric functions, such as periodicity and range, is crucial for finding all possible solutions. Practice with various methods and types of equations will improve problem-solving skills in this area.