Trigonometric Equations Solutions

Trigonometric Equations: These involve sine, cosine, and tangent functions to determine angles or lengths within geometrical scenarios.

Callout Definitions

• Cyclical Phenomena: Trigonometric equations are utilised to model cyclical phenomena, such as sound waves and alternating current (AC) circuits.
• General Solutions: These are expressions that summarise all possible solutions to trigonometric equations because of their periodic nature.
• Quadratic Trigonometric Equations: These equations involve trigonometric functions in a quadratic form and are essential for analysing wave patterns.

Basic Concepts and Definitions

• Trigonometric Equation: An equation that involves the trigonometric ratios of an unknown angle.
• General Solutions Importance: Trigonometric equations can provide infinite solutions due to periodicity.

Significance and Applications

• Sound Waves: Sine functions model frequency and pitch in sound waves.
• AC Circuit Design: Sine graphs represent alternating current circuits, illustrating voltage and current variations over time.

Techniques for Solving Trigonometric Equations

1. Simple Trigonometric Functions

• Common equations include:
  • $\sin x = \frac{1}{2}$
  • $\cos x = 0$
  • $\tan x = \sqrt{3}$
• Employ the unit circle to find solutions:

2. Inverse Trigonometric Functions

• Solve basic equations by using inverse functions, such as $x = \sin^{-1}(0.5)$.

3. Quadratic Trigonometric Equations

• Substitution Method: Replace trigonometric functions with algebraic variables for solving.
• Example: Convert $\sin^2 x$ into $u$ to solve $u^2 - u - 1 = 0$.

Variety of Trigonometric Identities

Trigonometric Identities: These equations demonstrate constant relationships among trigonometric functions.

Key Identities:

• Pythagorean Identities: e.g., $\sin^2x + \cos^2x = 1$.
• Angle Sum: $\sin(x \pm y) = \sin x \cos y \pm \cos x \sin y$.
• Double Angle: $\sin(2x) = 2\sin x \cos x$.

Step-by-Step Examples

Example 1: Solve $2\sin^2x + \cos^2x = 1$

1. Recall that $\sin^2x + \cos^2x = 1$, so $\sin^2x = 1 - \cos^2x$
2. Substitute this into our equation: $2(1 - \cos^2x) + \cos^2x = 1$
3. Simplify: $2 - 2\cos^2x + \cos^2x = 1$ $2 - \cos^2x = 1$ $-\cos^2x = -1$ $\cos^2x = 1$
4. Therefore, $\cos x = \pm 1$
5. Solutions: $x = 0, 2\pi, 4\pi, ...$ or $x = \pi, 3\pi, 5\pi, ...$
6. General solution: $x = n\pi$, where $n$ is an integer.

Graphical Representations

• Wave Diagram Interpretations:

Solving via Transformations

Function Transformations

• Amplitude Changes: These alter the vertical stretch of a graph.
• Period Adjustments: These changes modify how frequently the graph repeats.

Tips for Exam Preparation

• Graphing Tools: Regular practice with graphing tools enhances the visualisation of solutions.
• Common Mistakes: Be aware of neglecting negative solutions or entire cycles.