Trigonometric Identities Overview

Trigonometric identities and functions hold significant importance in Mathematics Advanced. These identities simplify expressions and aid in complex problem-solving across various fields.

Overview of Reciprocal Trigonometric Functions

Definitions:

• $\csc \, x$: Reciprocal of $\sin x$: $\csc x = \frac{1}{\sin x}$

  • Example: If $\sin x = \frac{1}{2}$, then $\csc x = 2$.

• $\sec \, x$: Reciprocal of $\cos x$: $\sec x = \frac{1}{\cos x}$

  • Example: If $\cos x = \frac{1}{2}$, then $\sec x = 2$.

• $\cot \, x$: Reciprocal of $\tan x$: $\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}$

  • Example: If $\tan x = 1$, then $\cot x = 1$.

Relationships with Primary Functions

• Expressions Derived from Functions:
  • Reciprocal identities such as $\csc x = \frac{1}{\sin x}$, $\sec x = \frac{1}{\cos x}$, and $\cot x = \frac{1}{\tan x}$ are essential for manipulating trigonometric equations.

Domain and Range

Function	Domain	Range
$\csc x$	$x \neq k\pi$	$(-\infty, -1] \cup [1, \infty)$
$\sec x$	$x \neq \frac{\pi}{2} + k\pi$	$(-\infty, -1] \cup [1, \infty)$
$\cot x$	$x \neq k\pi$	$(-\infty, \infty)$

Graphical Characteristics

• Periodicity and Asymptotes:

  • Asymptotes occur at undefined points, represented by vertical lines in graphs.

• Examples:

  • $y = \csc x$: Asymptotes at $x = n\pi$.
  • $y = \sec x$: Asymptotes at $x = \frac{(2n+1)\pi}{2}$.
  • $y = \cot x$: Asymptotes at $x = n\pi$.

Pythagorean Identities

Definitions and Derivations

• Pythagorean Identities:
  • $\cos^2x + \sin^2x = 1$
  • $1 + \tan^2x = \sec^2x$
  • $1 + \cot^2x = \csc^2x$

These identities simplify complex trigonometric expressions.

Derivation:

• From the unit circle, recognise that the Pythagorean Theorem results in $\cos^2 x + \sin^2 x = 1$.
• Adjust by dividing through to obtain $1 + \tan^2 x = \sec^2 x$ and $1 + \cot^2 x = \csc^2 x$.

Rationalising Trigonometric Ratios

Key Concepts

• Tangent Function:
  • $\tan x = \frac{\sin x}{\cos x}$ requires $\cos x \neq 0$ to avoid undefined expressions.

Examples and Simplification

• Simplify complex expressions such as $\frac{3\tan x + 2}{5\tan x - 1}$ by verifying $\cos x \neq 0$.
• Rationalise expressions accurately under these conditions.

Practice Questions with Solutions

Question 1: Verify the identity $\sec^2 x - \tan^2 x = 1$.

Solution: $\sec^2 x - \tan^2 x = \frac{1}{\cos^2 x} - \frac{\sin^2 x}{\cos^2 x} = \frac{1 - \sin^2 x}{\cos^2 x} = \frac{\cos^2 x}{\cos^2 x} = 1$

Question 2: Given $0 \leq x < 2\pi$, find when $\csc x = \sqrt{2}$.

Solution: $\csc x = \sqrt{2}$ $\frac{1}{\sin x} = \sqrt{2}$ $\sin x = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

This occurs when $x = \frac{\pi}{4}$ or $x = \frac{3\pi}{4}$ in the first two quadrants, and $x = \frac{5\pi}{4}$ or $x = \frac{7\pi}{4}$ in the third and fourth quadrants.

Evaluating Trigonometric Expressions

Reference and Complementary Angles

• Reference Angles standardise to 0°-90° by using quadrant relationships.
• Complementary Relationships between sine and cosine simplify evaluations, e.g., $\sin(30^\circ) = \cos(60^\circ)$.

Impact of Quadrant and Sign

• Sign determination by quadrant:
  • Quadrant I: All functions positive.
  • Quadrant II: Sine positive.
  • Quadrant III: Tangent positive.
  • Quadrant IV: Cosine positive.

Common Mistakes

• Incorrectly determining signs or choosing reference angles can affect accuracy.

Conclusion

Consistent practice transforms challenging trigonometric problems into manageable exercises, which is vital for exams and practical applications in fields such as engineering and physics.