Solving Trig Equations Revision Notes for Leaving Cert Mathematics

Solving Trig Equations

Overview

Solving trigonometric equations involves finding all values of the variable $x$ that satisfy a given equation within a specified domain, often using trigonometric identities, inverse functions, and symmetry properties of trigonometric graphs.

General Steps for Solving Trigonometric Equations

Step 1: Isolate the Trigonometric Function:

Rearrange the equation to express one trigonometric function in terms of $x$

Step 2: Apply Inverse Functions or Known Values:

Solve for $x$ using known values of trigonometric functions or inverse functions $(\sin^{-1}, \cos^{-1}, \tan^{-1})$ .

Step 3: Account for Periodicity:

Include all possible solutions by adding or subtracting multiples of the period of the trigonometric function.

Step 4: Use Identities as Needed:

Simplify complex expressions using identities like $\sin^2x + \cos^2x = 1$ , or product-to-sum formulas.

Step 5: Verify Solutions:

Check solutions within the specified domain to ensure validity.

Worked Examples

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Example 1: Solving a Basic Equation

Problem: Solve $\sin x = \frac{1}{2}$ for $x$ in $[0, 2\pi]$

Solution:

Step 1: Use the unit circle to find angles where $\sin x = \frac{1}{2}$

x = \frac{\pi}{6}, \frac{5\pi}{6}

Step 2: Verify these solutions are in the domain $[0, 2\pi]$

Answer: $x = \frac{\pi}{6}, \frac{5\pi}{6}$

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Example 2: Solving a Quadratic Trigonometric Equation

Problem: Solve $2\cos^2x - \cos x - 1 = 0$ for $x$ in $[0, 2\pi]$

Solution:

Step 1: Factorise:

(2\cos x + 1)(\cos x - 1) = 0

$\cos x = -\frac{1}{2}$ or $\cos x = 1$

Step 2: Solve each equation:

$\cos x = -\frac{1}{2}$ :

x = \frac{2\pi}{3}, \frac{4\pi}{3}

$\cos x = 1$ :

x = 0

Step 3: Combine solutions within the domain:

x = 0, \frac{2\pi}{3}, \frac{4\pi}{3}

Answer: $x = 0, \frac{2\pi}{3}, \frac{4\pi}{3}$

Summary

Solving trigonometric equations involves isolating the function, using periodicity, and applying trigonometric identities.
Verify solutions within the given domain.
Common tools include inverse functions, the unit circle, and trigonometric identities. These techniques are essential for solving trigonometric problems in geometry, physics, and engineering contexts.

Solving Trig Equations (Leaving Cert Mathematics): Revision Notes