Reciprocal Trigonometric Ratios

Reciprocal trigonometric ratios extend the basic trigonometric functions (\(\sin\), \(\cos\), and \(\tan\)) by defining their reciprocals: \(\sec\theta\), \(\csc\theta\), and \(\cot\theta\). These ratios are essential in understanding advanced trigonometric identities and their applications.

Key Concepts

• Definitions: The reciprocal trigonometric ratios are defined as:
  • \(\sec\theta = \frac{1}{\cos\theta}\), valid for \(\cos\theta \neq 0\).
  • \(\csc\theta = \frac{1}{\sin\theta}\), valid for \(\sin\theta \neq 0\).
  • \(\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}\), valid for \(\sin\theta \neq 0\).
• Pythagorean Identities: Reciprocal trigonometric ratios satisfy these key identities:
  • \(1 + \tan^2\theta = \sec^2\theta\).
  • \(1 + \cot^2\theta = \csc^2\theta\).
• Inverse Trigonometric Functions: The inverse functions allow calculation of angles from given trigonometric values:
  • \(f(x) = \arcsin x\), domain: \([-1, 1]\), range: \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
  • \(f(x) = \arccos x\), domain: \([-1, 1]\), range: \([0, \pi]\).
  • \(f(x) = \arctan x\), domain: \(\mathbb{R}\), range: \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Examples

• Example 1: Find \(\sec\theta\) and \(\cot\theta\) given \(\sin\theta = \frac{3}{5}\), \(\theta\) in the first quadrant.
  • Step 1: Use \(\sin^2\theta + \cos^2\theta = 1\) to find \(\cos\theta = \frac{4}{5}\).
  • Step 2: Calculate \(\sec\theta = \frac{1}{\cos\theta} = \frac{5}{4}\).
  • Step 3: Calculate \(\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{4}{3}\).
• Example 2: Solve \(1 + \cot^2\theta = \csc^2\theta\) to verify values.
  • Step 1: Substitute \(\cot^2\theta = \frac{\cos^2\theta}{\sin^2\theta}\) and \(\csc^2\theta = \frac{1}{\sin^2\theta}\).
  • Step 2: Verify \(1 + \frac{\cos^2\theta}{\sin^2\theta} = \frac{1}{\sin^2\theta}\).
  • Step 3: Simplify to show \(1 = \sin^2\theta + \cos^2\theta\), which holds true.