Trigonometric Equations

A trigonometric equation is an equation that involves one or more trigonometric functions of a variable. Solving a trigonometric equation means finding all the angles that satisfy the equation. Because trigonometric functions are periodic, these equations typically have an infinite number of solutions unless the domain is restricted.

General Solutions vs. Particular Solutions

Particular Solutions

Solutions that lie within a specific restricted interval, usually [0,2π)[0, 2\pi) or [0∘,360∘)[0^\circ, 360^\circ).

General Solutions

Formulas that represent all possible solutions across the entire domain (−∞,∞)(-\infty, \infty) by adding integer multiples of the function's period.
  • For sin⁡θ\sin \theta and cos⁡θ\cos \theta, add 2nπ2n\pi (where nn is an integer).
  • For tan⁡θ\tan \theta, add nπn\pi.
Key Takeaways
  • Periodic Nature: Always pay attention to whether a problem asks for solutions within a specific interval (e.g., [0,2π)[0, 2\pi)) or for the general solution.

Procedure

  1. Isolate the Function: Use algebraic techniques to isolate the trigonometric function (e.g., get sin⁡x\sin x alone on one side).
  2. Find the Reference Angle: Determine the positive acute reference angle in Quadrant I using the inverse trigonometric function on the absolute value of the ratio.
  3. Determine Quadrants: Use the ASTC rule to determine which quadrants the solutions lie in, based on the sign of the isolated ratio.
  4. Find Principal Angles: Calculate the angles in the identified quadrants.
  5. Add the Period: For a general solution, append +2nπ+ 2n\pi for sine/cosine or +nπ+ n\pi for tangent (where n∈Zn \in \mathbb{Z}).

Extraneous Solutions

Identifying Extraneous Solutions

When solving trigonometric equations, squaring both sides or multiplying by a variable expression can introduce extraneous solutions—values that mathematically solve the transformed equation but do not satisfy the original equation.
Note: Always plug your final answers back into the original equation to verify their validity, especially if your solution process involved squaring.

Solving Basic Linear Trigonometric Equations

To solve basic linear equations, isolate the trigonometric function on one side of the equation, and then use the inverse trigonometric function or knowledge of the unit circle to find the angle.
Key Takeaways
  • Isolation: Treat the trigonometric function like an algebraic variable (e.g., u=cos⁡xu = \cos x) and solve for it first.
  • Quadrants: After isolating the function, determine the reference angle and use the ASTC rule to find all valid angles in the specified domain.

Solving Quadratic Trigonometric Equations

Trigonometric equations that resemble quadratic forms can be solved by factoring, using the quadratic formula, or extracting square roots. Often, you will need to use Pythagorean identities to ensure the equation contains only one type of trigonometric function.

Note

If an equation contains mixed functions like sin⁡2x\sin^2 x and cos⁡x\cos x, use the identity sin⁡2x=1−cos⁡2x\sin^2 x = 1 - \cos^2 x to rewrite the equation entirely in terms of cosine before factoring.
Key Takeaways
  • Substitution: Using an algebraic substitution like u=sin⁡xu = \sin x can make factoring quadratic trigonometric equations much easier to visualize.
  • Identities: Use Pythagorean identities to ensure the quadratic equation is written in terms of a single trigonometric function.
  • Extraneous Solutions: If you divide both sides by a trigonometric expression, or square both sides, always verify your final answers as extraneous solutions may be introduced.

Equations of the Form Asin⁡x+Bcos⁡x=CA \sin x + B \cos x = C

Linear combinations of sine and cosine can be solved elegantly without squaring both sides (which introduces extraneous solutions). The most robust method is to use the Harmonic Addition theorem to combine the two terms.

Procedure

  • Find the Amplitude: Calculate R=A2+B2R = \sqrt{A^2 + B^2}.
  • Find the Phase Angle: Find α\alpha such that cos⁡α=AR\cos \alpha = \frac{A}{R} and sin⁡α=BR\sin \alpha = \frac{B}{R} (or tan⁡α=BA\tan \alpha = \frac{B}{A}).
  • Rewrite the Equation: Substitute the left side with Rsin⁡(x+α)=CR \sin(x + \alpha) = C.
  • Solve for xx: If ∣C∣≤R|C| \le R, the equation has real solutions. Divide by RR and use arcsin: sin⁡(x+α)=CR\sin(x + \alpha) = \frac{C}{R}. Determine the values for x+αx + \alpha in the valid domain, then subtract α\alpha to find xx.

Phasors in AC Circuits

The Harmonic Addition Theorem is the mathematical foundation for Phasor Analysis in electrical engineering. In alternating current (AC) circuits, voltages and currents are often represented as sinusoidal waves with the same frequency but different amplitudes and phases: Asin⁡(ωt)+Bcos⁡(ωt)A \sin(\omega t) + B \cos(\omega t).
Instead of constantly applying trigonometric identities to add these waves, engineers convert them into "phasors" (complex numbers representing magnitude RR and phase angle α\alpha). The addition of sine waves becomes simple vector addition in the complex plane, vastly simplifying the analysis of RLC circuits.
Key Takeaways
  • Linear Combinations: Avoid squaring Asin⁡x+Bcos⁡x=CA \sin x + B \cos x = C. Instead, rewrite it as a single sine or cosine function using R=A2+B2R = \sqrt{A^2 + B^2}.

Equations with Multiple Angles

When the argument of the trigonometric function is a multiple of xx (e.g., sin⁡(2x)\sin(2x), cos⁡(3x)\cos(3x)), you must first find the solutions for the entire argument, and then divide to solve for xx. This process often requires finding solutions beyond the standard [0,2π)[0, 2\pi) interval before dividing.
Key Takeaways
  • Domain Expansion: For multiple angle equations like sin⁡(kx)=c\sin(kx) = c, expand the solution domain to [0,k⋅2π)[0, k \cdot 2\pi) to find all possible values for kxkx before dividing by kk.
  • Double Angle Identities: Sometimes, it's easier to use double angle identities (e.g., sin⁡(2x)=2sin⁡xcos⁡x\sin(2x) = 2\sin x \cos x) to convert the equation back to single angles, factor, and solve.
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