Understanding how to solve trigonometric equations is a crucial part of the Cambridge IGCSE mathematics syllabus. This section focuses on solving equations such as $\sin x = \frac{2}{3}$ within the domain $0° ≤ x ≤ 360°$ and utilising inverse trigonometric functions to find solutions. Trigonometric equations are foundational in various fields, including engineering, physics, and geometry, making their mastery important for students.

Introduction to Trigonometric Equations

Trigonometric equations involve trigonometric functions (sine, cosine, tangent, etc.) with variables. The goal is to find the values of these variables that satisfy the given equation. These equations can often have multiple solutions, especially within a given range such as $0°$ to $360°$.

Inverse Trigonometric Functions

Before diving into solving trigonometric equations, it's essential to understand inverse trigonometric functions. These functions, denoted as $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$, are used to find the angle that corresponds to a given trigonometric value.

• Definition: Inverse trigonometric functions allow us to find the angle whose trigonometric function gives a specific value.
• Domain and Range: The range of $\sin^{-1}$ and $\cos^{-1}$ is $-90° ≤ y ≤ 90°$ and $0° ≤ y ≤ 180°$ respectively, while $\tan^{-1}$ has a range of -90° < y < 90°.

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Solving Equations with Sine

Solving equations involving the sine function requires understanding its periodic nature and how it behaves over the interval $0°$ to $360°$.

Example 1: Solve $\sin x = \frac{2}{3}$ for $0° ≤ x ≤ 360°$

1. Use the Inverse Function: First, calculate the angle whose sine is $\frac{2}{3}$ using $x = \sin^{-1}(\frac{2}{3})$.

2. The Principal Solution: The principal solution is the smallest angle that satisfies the equation. This can be found using a calculator.

3. Identify Additional Solutions: Since the sine function is periodic, other solutions can be found by considering the symmetry of the sine curve. The other solution in the range $0°$ to $360°$ is $180° - x$.

Calculation:

• Principal Solution: Calculating the principal solution using $\sin^{-1}(\frac{2}{3})$ yields approximately $41.8°$.
• Additional Solution: The additional solution is $180° - 41.8° = 138.2°$.

Therefore, the solutions for $\sin x = \frac{2}{3}$ within $0°$ to $360°$ are $x \approx 41.8°$ and $x \approx 138.2°$.

Solving Equations with Cosine

Similar to sine, solving equations with the cosine function involves understanding its unique properties, such as its periodicity and symmetry about the $y$-axis.

Example 2: Solve $\cos x = \frac{1}{2}$ for $0° ≤ x ≤ 360°$

1. Use the Inverse Function: Calculate the angle with $x = \cos^{-1}(\frac{1}{2})$.

2. Principal Solution: This yields $60°$, a principal solution.

3. Identify Additional Solutions: Given the symmetry of the cosine curve, another solution in the given range is $360° - 60° = 300°$.

Calculation:

• Principal Solution: $\cos^{-1}(\frac{1}{2}) = 60°$.
• Additional Solution: $360° - 60° = 300°$.

Thus, the solutions for $\cos x = \frac{1}{2}$ within $0°$ to $360°$ are $60°$ and $300°$.

Solving Equations with Tangent

The tangent function has a period of $180°$, making its solutions slightly different to identify.

Example 3: Solve $\tan x = 1$ for $0° ≤ x ≤ 360°$

1. Use the Inverse Function: Start with $x = \tan^{-1}(1)$.

2. Principal Solution: This calculation gives $45°$.

3. Periodicity of Tangent: Since tangent has a period of $180°$, the other solution is $45° + 180° = 225°$.

Calculation:

• Principal Solution: $\tan^{-1}(1) = 45°$.
• Additional Solution: $225°$.

The solutions for $\tan x = 1$ within $0°$ to $360°$ are $45°$ and $225°$.

Key Points to Remember

• Inverse Functions: Crucial for starting the process of finding solutions.
• Symmetry and Periodicity: Essential for identifying all possible solutions within a given range.
• Practice: The best way to master solving trigonometric equations is through consistent practice with a variety of problems.