Trigonometric Equations (Edexcel A-Level Mathematics): Revision Notes

5.3.4 Strategy for Trigonometric Equations

Solving trigonometric equations can be challenging, but with a systematic approach, you can tackle them efficiently. Here's a general strategy to help you solve a wide range of trigonometric equations:

1. Understand the Equation

• Identify which trigonometric function(s) are involved (e.g., $\sin \theta, \cos \theta, \tan \theta$).
• Determine whether the equation is linear (e.g.,$\sin \theta = \frac{1}{2}$), quadratic (e.g., $2\sin^2 \theta - \sin \theta - 1 = 0$), or involves multiple trigonometric functions.

2. Simplify the Equation

• Isolate the trigonometric function: Get the trigonometric function (e.g.,$\sin \theta, \cos \theta$) by itself on one side of the equation.
• Factor or Simplify: If the equation is quadratic or more complex, try factoring, expanding, or simplifying using trigonometric identities.
• Substitution: If multiple trigonometric functions are involved (e.g., $\sin^2 \theta + \cos^2 \theta = 1$), consider substituting using identities to reduce the equation to one trigonometric function.

3. Use Trigonometric Identities

• Apply identities such as the Pythagorean identities, double-angle identities, sum-to-product identities, or half-angle identities to simplify the equation.
• Common identities include: $\sin^2 \theta + \cos^2 \theta = 1$ $\tan \theta = \frac{\sin \theta}{\cos \theta}$ $\sin(2\theta) = 2\sin \theta \cos \theta$

4. Solve the Simplified Equation

• Linear Equations:
  • Solve for the angle by taking the inverse trigonometric function (e.g.,$\sin^{-1}, \cos^{-1}, \tan^{-1}$).
  • Consider all solutions within the given interval by accounting for the periodic nature of the trigonometric functions.
• Quadratic Equations:
  • Factor if possible, or use the quadratic formula: $u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Back-substitute to solve for the angle.
  • Check for extraneous solutions, especially if you square both sides of the equation.

5. Consider the General Solution

• Trigonometric functions are periodic, meaning they repeat their values at regular intervals.
• For $\sin \theta = k and \cos \theta = k:$
  • $\theta = \theta_0 + 360^\circ n\ (or\ 2\pi$ n) where n is any integer.
  • Include both primary solutions $\theta_0 and 180^\circ - \theta_0 (or\ \pi - \theta_0).$
• For $\tan \theta = k:$
  • $\theta = \theta_0 + 180^\circ n (or\ \pi n).$

6. List All Solutions in the Given Interval

• Ensure that all solutions are within the specified interval, typically $0^\circ to\ 360^\circ$ or $0\ to\ 2\pi$ radians.
• Adjust the general solution accordingly to fit within this range.

7. Verify Solutions

• Substitute each solution back into the original equation to ensure it satisfies the equation.
• Discard any extraneous solutions that might have arisen from squaring both sides or other manipulations.

Summary:

• Simplify the trigonometric equation using identities or substitution.
• Solve the resulting equation for the variable.
• Consider the periodic nature of trigonometric functions to find all solutions within the given interval.
• Verify the solutions to ensure correctness.

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Solving Trig Equations Involving Compound Angles

By "compound angle," we mean "an angle more complicated than just $\theta.$"

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