Trigonometric Equations: Change of Variable

Trigonometric equations are equations that involve trigonometric functions such as sine, cosine, tangent, etc. Solving these equations often requires the use of various algebraic and trigonometric techniques. One such technique is the change of variable, which can simplify complex trigonometric equations into more manageable forms.

Understanding Change of Variable

Change of variable is a method where we substitute a trigonometric expression with a single variable, typically denoted as $t$, $u$, or another letter. This substitution can transform a trigonometric equation into a polynomial or a simpler trigonometric equation, which is often easier to solve.

Why Use Change of Variable?

• Simplification: It can turn a complex equation into a simpler one.
• Familiar Forms: It allows us to work with polynomial-like equations, which are more familiar to many students.
• Reduces Complexity: It can reduce the number of trigonometric functions in an equation, making it less complex.
• Solves Higher Degree Equations: It is particularly useful for solving higher degree trigonometric equations.

Common Substitutions

Here are some common substitutions used in the change of variable method:

Trigonometric Expression	Substitution	Resulting Variable
$\sin(x)$	$t = \sin(x)$	$t$
$\cos(x)$	$t = \cos(x)$	$t$
$\tan(x)$	$t = \tan(x)$	$t$
$\sin^2(x)$	$t = \sin^2(x)$	$t$
$\cos^2(x)$	$t = \cos^2(x)$	$t$
$\tan^2(x)$	$t = \tan^2(x)$	$t$
$2\sin(x)\cos(x)$	$t = \sin(2x)$	$t$

Steps for Change of Variable

1. Identify the Substitution: Look for trigonometric expressions that can be replaced with a single variable.
2. Make the Substitution: Replace the identified expression(s) with the new variable.
3. Solve the New Equation: Solve the resulting equation as you would with any algebraic equation.
4. Back-Substitute: Once you have the solution for the new variable, substitute back to get the solution in terms of the original trigonometric function.
5. Check for Extraneous Solutions: Verify that the solutions satisfy the original equation, as the process may introduce extraneous solutions.

Formulas Involving Change of Variable

When changing variables, we often use trigonometric identities to assist in the substitution. Some useful identities include:

• Pythagorean Identity: $\sin^2(x) + \cos^2(x) = 1$
• Double Angle Formulas: $\sin(2x) = 2\sin(x)\cos(x)$, $\cos(2x) = \cos^2(x) - \sin^2(x)$
• Half Angle Formulas: $\sin^2(x) = \frac{1 - \cos(2x)}{2}$, $\cos^2(x) = \frac{1 + \cos(2x)}{2}$

Examples

Example 1: Solving a Quadratic Trigonometric Equation

Consider the equation $\sin^2(x) - \sin(x) - 2 = 0$.

1. Identify the Substitution: Let $t = \sin(x)$.
2. Make the Substitution: The equation becomes $t^2 - t - 2 = 0$.
3. Solve the New Equation: Factor to get $(t - 2)(t + 1) = 0$, so $t = 2$ or $t = -1$.
4. Back-Substitute: Since $t = \sin(x)$, we have $\sin(x) = 2$ (which has no solution) and $\sin(x) = -1$.
5. Check for Extraneous Solutions: $\sin(x) = -1$ gives $x = \frac{3\pi}{2} + 2k\pi$, where $k$ is an integer.

Example 2: Solving a Trigonometric Equation with Multiple Functions

Consider the equation $2\sin(x)\cos(x) + \cos(x) = 0$.

1. Identify the Substitution: Let $t = \sin(x)$ and note that $2\sin(x)\cos(x) = \sin(2x)$.
2. Make the Substitution: The equation becomes $\sin(2x) + \cos(x) = 0$. Using the double angle formula, we get $2t\cos(x) + \cos(x) = 0$.
3. Solve the New Equation: Factor to get $\cos(x)(2t + 1) = 0$, so $\cos(x) = 0$ or $t = -\frac{1}{2}$.
4. Back-Substitute: For $\cos(x) = 0$, $x = \frac{\pi}{2} + k\pi$. For $t = -\frac{1}{2}$, $\sin(x) = -\frac{1}{2}$, which gives $x = \frac{7\pi}{6} + 2k\pi$ or $x = \frac{11\pi}{6} + 2k\pi$.
5. Check for Extraneous Solutions: Verify that each solution satisfies the original equation.

By using the change of variable technique, we can effectively solve trigonometric equations that might otherwise be difficult to handle. Practice with various trigonometric equations is essential to become proficient in this method.