The Main Idea

Equations that involve only one trigonometric function can be solved much like basic algebraic equations—first isolate the trig function, then use inverse trig or the unit circle to find solutions. Because trig functions are periodic, solutions often come in infinite families. The key is to solve for a “base” angle and then add multiples of the period.

Quick Tips: Solving Single-Trig Equations

1. Isolate the Function

2. Find the Reference Angle

   • Use special triangles, the unit circle, or inverse trig to get the principal angle.

3. Determine All Solutions in One Cycle

   • Check which quadrants have the same sign for the trig function.

4. Add the General Solution

   • Sine and cosine repeat every [latex]2\pi[/latex].

   • Tangent repeats every [latex]\pi[/latex].

The Main Idea

Some trigonometric equations can be written in forms that allow factoring, much like algebraic quadratics. Once factored, each piece can be set equal to zero, and the resulting simpler trig equations are solved individually. Because trig functions are periodic, each factor can produce multiple solutions across all cycles.

Quick Tips: Solving by Factoring

1. Set the Equation to Zero

   • Rearrange so one side equals zero.

2. Factor the Expression

   • Factor out common terms or use quadratic-style factoring.

3. Solve Each Equation

   • Set each factor equal to zero:

   • Use the unit circle or inverse trig to find solutions.

4. Add the General Solutions

   • Since sine and cosine have a period of [latex]2\pi[/latex], add [latex]\pm 2\pin[/latex] to the end of each solution.

   • Since tangent has a period of [latex]\pi[/latex], add [latex]\pm \pi[/latex] to each unique solution.

5. Check Solutions

   • Always substitute back into the original equation, especially when squaring or multiplying factors may have introduced extraneous answers.

The Main Idea

Some trigonometric equations cannot be solved directly but become manageable when you apply the fundamental identities. Substituting reciprocal, quotient, or Pythagorean identities can reduce an equation to one involving a single trig function. Once simplified, the equation can be solved using standard methods (isolate, find reference angle, check quadrants, and add general solutions).

Quick Tips: Solving with Identities

1. Look for Opportunities to Substitute

   • Replace quotients: [latex]\tan\theta = \dfrac{\sin\theta}{\cos\theta}[/latex], [latex]\cot\theta = \dfrac{\cos\theta}{\sin\theta}[/latex].

   • Replace reciprocals: [latex]\sec\theta = \dfrac{1}{\cos\theta}[/latex], [latex]\csc\theta = \dfrac{1}{\sin\theta}[/latex].

   • Use Pythagorean identities:

     • [latex]\sin^{2}\theta + \cos^{2}\theta = 1[/latex]

     • [latex]1 + \tan^{2}\theta = \sec^{2}\theta[/latex]

     • [latex]1 + \cot^{2}\theta = \csc^{2}\theta[/latex]

2. Convert to a Single Function

   • Try to rewrite the equation so it involves only sine, only cosine, or only tangent.

3. Check Quadrants and Ranges

   • Make sure your answers are in the correct quadrants for the trig function’s sign.

4. Be Alert for Extraneous Solutions

   • Identities sometimes introduce undefined values (like dividing by [latex]\cos\theta[/latex] when [latex]\cos\theta = 0[/latex]). Always verify in the original equation.

The Main Idea

Trigonometric equations sometimes involve multiples of the variable, like [latex]\sin(2\theta)[/latex] or [latex]\cos(3\theta)[/latex]. To solve these, we treat the inside of the trig function as a single variable first, solve for that angle, and then divide out the multiple to find all possible solutions. Because of periodicity, multiple angles produce more solutions within a single cycle, so it’s important to capture every possibility.

Quick Tips: Solving Multiple-Angle Equations

1. Isolate the Function
2. Let a Temporary Variable Stand In
3. Solve for All Angles in One Cycle
4. Return to the Original Variable