5.7.1 Strategy for Further Trigonometric Equations

Solving further trigonometric equations often involves more complex strategies than basic trigonometric equations. These equations might involve multiple angles, compound angles, or require the use of identities like double-angle or sum-to-product identities. Here's a strategy to approach and solve these types of equations:

1. Understand the Equation

• Identify which trigonometric functions and identities are involved (e.g., sine, cosine, tangent, double angles, sum/difference of angles).
• Determine if the equation involves compound angles, multiple angles, or identities that can be simplified.

2. Simplify Using Trigonometric Identities

• Expand or Factor: If the equation involves compound angles (e.g., $\sin(A + B)$), use the sum/difference identities to expand them.
• Use Double-Angle or Half-Angle Identities: If the equation involves double angles (e.g., $\sin 2\theta$), apply the double-angle identities: $\sin 2\theta = 2\sin \theta \cos \theta, \quad \cos 2\theta = 2\cos^2 \theta - 1 \text{ or } 1 - 2\sin^2 \theta$
• Convert Products to Sums: If the equation involves products of trigonometric functions, use product-to-sum identities: $\sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)]$
• Simplify Complex Fractions: If the equation has fractions, multiply both sides by a common denominator to simplify.

3. Isolate the Trigonometric Function

• Isolate: Try to isolate one trigonometric function on one side of the equation. For example, if you have: $2 \sin \theta + \sqrt{3} = 0$ Isolate $\ \sin \theta \ by\ subtracting \ \sqrt{3}$ from both sides and dividing by $2$.

4. Use Substitution If Necessary

• If the equation is quadratic in form, or involves multiple trigonometric terms (e.g., $\ \sin^2 \theta$) and ( $\sin \theta$), use substitution:
  • Let $u = \sin \theta \ (\ or \ \cos \theta)$, solve the resulting quadratic equation for $\ u$, then back-substitute to find $\theta .$

5. Consider All Possible Solutions

• General Solution: Trigonometric functions are periodic, so after solving for the basic angle, consider the general solution:
  • For $\ \sin \theta = k \ or \ \cos \theta = k :$ $\theta = \theta_0 + 360^\circ n \quad \text{or} \quad \theta = 180^\circ - \theta_0 + 360^\circ n$
  • For $\tan \theta = k :$ $\theta = \theta_0 + 180^\circ n$
• Specific Interval: If the problem specifies an interval (e.g., $0^\circ \leq \theta \leq 360^\circ$), ensure that all solutions fall within this interval.

6. Check for Extraneous Solutions

• After solving, substitute your solutions back into the original equation to ensure they satisfy it.
• Extraneous solutions often arise when you square both sides of an equation or use an identity that introduces additional roots.

7. Special Cases and Techniques

• Using Sum and Difference Identities: If the equation involves terms like $\sin(A \pm B) ,$expand them using: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
• Using $R$ Addition Formulae: If the equation is of the form$\ a \cos \theta + b \sin \theta = c ,$consider using the R addition formula: $R \cos(\theta \pm \alpha) = a \cos \theta + b \sin \theta$ Where $\ R = \sqrt{a^2 + b^2} \ and \tan \alpha = \frac{b}{a} .$

Example Problems:

Summary: