How to Use Basic Techniques for Solving Trigonometric Equations

A Step-by-step Guide to Using Basic Techniques for Solving Trigonometric Equations

Here are the steps to solve trigonometric equations:

Step 1: Simplify the Equation

Often, trigonometric equations will involve more than one function (like \(sin, cos\), and \(tan\)) or complicated expressions. The first step is always to simplify the equation as much as possible. This could involve using trigonometric identities (like \(sin^2x + cos^2x = 1\) or \(sin(2x) = 2sin(x)cos(x)\)) to express everything in terms of one function or expanding out expressions to simplify them.

For example, if you have the equation \(2sin(x)cos(x) = sin(x)\), you could use the identity \(sin(2x) = 2sin(x)cos(x)\) to simplify the equation to \(sin(2x) = sin(x)\).

Step 2: Solve for the Trigonometric Function

Once you’ve simplified the equation, the next step is to isolate the trigonometric function on one side of the equation. This might involve adding or subtracting terms from both sides, multiplying or dividing both sides by a constant, or using algebraic techniques to isolate the function.

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Continuing the previous example, \(sin(2x) = sin(x)\) can be rewritten as \(sin(2x) – sin(x) = 0\).

Step 3: Use Inverse Trigonometric Functions

Once you’ve isolated the trigonometric function, the next step is to use an inverse trigonometric function to solve for the variable. The inverse trigonometric functions are arcsin, arccos, and arctan. These functions “undo” the trigonometric functions: for example, if \(sin(x) = y\), then \(arcsin(y) = x\).

Continuing the previous example, \(sin(2x) – sin(x) = 0\) could be solved by setting each term to zero (since \(a\times b=0\) implies \(a=0\) or \(b=0\)), giving \(sin(2x) = 0\) and \(sin(x) = 0\). Using arcsin on both equations, you’d get \(x = arcsin(0)\) and \(2x = arcsin(0)\).

Step 4: Solve for the Variable

Once you’ve used the inverse trigonometric function, you should have an equation that you can solve for the variable. This might involve additional algebraic manipulation.

In our previous example, \(arcsin(0) = 0\), so \(x = 0\) is a solution. For \(2x = arcsin(0)\), we divide by \(2\) to get \(x = 0\).

Step 5: Check for Additional Solutions

For trigonometric equations, it’s important to remember that there can be more than one solution since the trigonometric functions are periodic. This means that they repeat their values at regular intervals. For the \(sin, cos\), and \(tan\) functions, this interval is \(2π\).

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For the equation \(sin(x) = 0, x = 0\) is a solution, but so is \(x = π, x = 2π, x = -π\), etc.

Step 6: Validate Your Solution

Finally, substitute your solutions back into the original equation to make sure they are correct.

Please remember, the steps might vary depending on the complexity and type of the trigonometric equation. The steps mentioned above are a generalized approach.