Trigonometry (Pure Mathematics 1 - 9709)

1. Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the ratios of its sides. For a right-angled triangle with angle $\theta$:

• Sine (sin): $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine (cos): $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent (tan): $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$
• Cosecant (csc): $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite}}$
• Secant (sec): $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}}$
• Cotangent (cot): $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}}$

These functions are defined for angles between 0 and $2\pi$ (or 0 and 360 degrees).

2. Trigonometric Identities

Trigonometric identities are equations that are always true for certain values of the angle. Some key identities include:

• Pythagorean Identity: $\sin^2(\theta) + \cos^2(\theta) = 1$
• Double Angle Identities:
  • $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
  • $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta)$
• Sum and Difference Identities:
  • $\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)$
  • $\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)$
  • $\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)}$
• Product to Sum Identities:
  • $\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)$
  • $\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)$

3. Trigonometric Equations

Solving trigonometric equations involves finding the values of $\theta$ that satisfy the equation. This often requires using the properties of trigonometric functions and identities.

Example: Solve $\sin(\theta) = 0.5$ for $0 \le \theta < 2\pi$.

We know that $\sin(30^\circ) = 0.5$. The sine function is positive in the first and second quadrants. Therefore, the solutions are:

• $\theta = \pi/6$ (first quadrant)
• $\theta = \pi - \pi/6 = 5\pi/6$ (second quadrant)

4. Solutions to Trigonometric Equations

To find all solutions to a trigonometric equation, consider the period of the trigonometric function. For example, the period of $\sin(\theta)$ is $2\pi$. Therefore, the general solutions for $\sin(\theta) = 0.5$ are:

$\theta = \frac{\pi}{6} + 2\pi n$ and $\theta = \frac{5\pi}{6} + 2\pi n$, where $n$ is an integer.

5. Graphs of Trigonometric Functions

The graphs of trigonometric functions have specific characteristics:

Function	Period	Amplitude	Key Features
$y = \sin(\theta)$	$2\pi$	1	Starts at the origin, oscillates between -1 and 1.
$y = \cos(\theta)$	$2\pi$	1	Starts at the maximum value, oscillates between -1 and 1.
$y = \tan(\theta)$	$\pi$	Undefined	Has vertical asymptotes where $\cos(\theta) = 0$ (i.e., $\theta = \pi/2, 3\pi/2, ...$).

The general form of a trigonometric function is $y = A\sin(\theta + \phi) + C$, where:

• $|A|$ is the amplitude
• $\phi$ is the phase shift
• $C$ is the vertical shift