Trigonometric equations, integral to A-Level Pure Mathematics, involve the manipulation and solution of equations containing trigonometric functions. These equations are pivotal in understanding various mathematical phenomena, particularly in fields like physics, engineering, and geometry. This section aims to provide a comprehensive guide to solving trigonometric equations, tailored for A-Level students.

Understanding and Applying Trigonometric Identities

Trigonometric identities are invaluable tools in mathematics for simplifying and evaluating complex expressions. Mastery of these identities is essential for solving a variety of trigonometry problems. This section covers key identities along with solutions to example problems for clear understanding.

Basic Trigonometric Identities

Pythagorean Identities

1. Identity: $(\cos \theta)^2 + (\sin \theta)^2 \equiv 1$

• Example: If $\sin \theta = \frac{3}{5}$, find $\cos \theta$.
• Solution: $\cos \theta = \pm \sqrt{1 - \left(\frac{3}{5}\right)^2} = \pm 0.8$

2. Identity: $1 + (\tan \theta)^2 \equiv (\sec \theta)^2$

• Example: Simplify $1 + \tan^2 45^\circ.$
• Solution: $1 + 1 = 2$

3. Identity: $(\cot \theta)^2 + 1 \equiv (\cosec \theta)^2$

• Example: Verify $\cot^2 45^\circ + 1 = \cosec^2 45^\circ$.
• Solution: $\left(\frac{1}{1}\right)^2+1=\left(\frac{\sqrt{2}}{1}\right)^2$, which simplifies to $2=2$, verifying the identity.

Double Angle Identities

1. Sin Double Angle

• Identity: $\sin 2A \equiv 2 \sin A \cos A$
  • Example: Find $\sin 2A$ when $\sin A = \frac{1}{2}$ and $\cos A = \frac{\sqrt{3}}{2}.$
    • Solution: $\sin 2A = 2 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}$

2. Cos Double Angle

• Identity: $\cos 2A \equiv (\cos A)^2 - (\sin A)^2$
  • Example: Express $\cos 60^\circ$ using the double angle identity.
    • Solution: $\cos 60^\circ = 2 \times \left(\frac{1}{2}\right)^2 - 1 = 0.$

3. Tan Double Angle

• Identity: $\tan 2A \equiv \frac{2 \tan A}{1 - (\tan A)^2}$
  • Example: Simplify $\tan 2A$ when $\tan A = 1$.
    • Solution: Undefined (due to division by zero)

Addition and Subtraction Identities

1. Sin Addition and Subtraction

• Identity: $\sin (A \pm B) \equiv \sin A \cos B \pm \cos A \sin B$
  • Example: Find $\sin (45^\circ + 30^\circ)$.
    • Solution: $\sin 75^\circ = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} \approx 0.966$

2. Cos Addition and Subtraction

• Identity: $\cos (A \pm B) \equiv \cos A \cos B \mp \sin A \sin B$
  • Example: Calculate $\cos (60^\circ - 45^\circ)$.
    • Solution: $\cos 15^\circ = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} \approx 0.966$

3. Tan Addition and Subtraction

• Identity: $\tan (A \pm B) \equiv \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
  • Example: Simplify $\tan (45^\circ + 30^\circ)$.
    • Solution: $\tan 75^\circ = \frac{\sqrt{3}/3 + 1}{1 - \sqrt{3}/3} \approx 3.732$