Ratios & Identities: Trigonometric identities

Ratios & Identities: Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable where both sides of the equation are defined. These identities are useful in simplifying expressions and solving trigonometric equations.

Basic Trigonometric Ratios

The six basic trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). They are defined as follows for a right-angled triangle:

  • $\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}$
  • $\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$
  • $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\sin(\theta)}{\cos(\theta)}$
  • $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite side}}$
  • $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent side}}$
  • $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)} = \frac{\text{adjacent side}}{\text{opposite side}}$

Fundamental Trigonometric Identities

There are several fundamental trigonometric identities that are derived from the basic trigonometric ratios. Here are some of the most important ones:

Pythagorean Identities

These identities are derived from the Pythagorean theorem.

  1. $\sin^2(\theta) + \cos^2(\theta) = 1$
  2. $1 + \tan^2(\theta) = \sec^2(\theta)$
  3. $1 + \cot^2(\theta) = \csc^2(\theta)$

Reciprocal Identities

These identities express the reciprocal relationship between sine, cosine, and their respective reciprocals.

  1. $\sin(\theta) = \frac{1}{\csc(\theta)}$
  2. $\cos(\theta) = \frac{1}{\sec(\theta)}$
  3. $\tan(\theta) = \frac{1}{\cot(\theta)}$

Quotient Identities

These identities express tangent and cotangent in terms of sine and cosine.

  1. $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$
  2. $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$

Co-Function Identities

These identities relate the trigonometric functions of complementary angles.

  1. $\sin(\theta) = \cos(90^\circ - \theta)$
  2. $\cos(\theta) = \sin(90^\circ - \theta)$
  3. $\tan(\theta) = \cot(90^\circ - \theta)$
  4. $\csc(\theta) = \sec(90^\circ - \theta)$
  5. $\sec(\theta) = \csc(90^\circ - \theta)$
  6. $\cot(\theta) = \tan(90^\circ - \theta)$

Even-Odd Identities

These identities express the symmetry properties of the trigonometric functions.

  1. $\sin(-\theta) = -\sin(\theta)$
  2. $\cos(-\theta) = \cos(\theta)$
  3. $\tan(-\theta) = -\tan(\theta)$
  4. $\csc(-\theta) = -\csc(\theta)$
  5. $\sec(-\theta) = \sec(\theta)$
  6. $\cot(-\theta) = -\cot(\theta)$

Examples

Here are some examples to illustrate the use of trigonometric identities:

Example 1: Simplifying an Expression

Simplify the expression $\sin^2(\theta) + \cos^2(\theta)$.

Solution:

Using the Pythagorean identity, we know that $\sin^2(\theta) + \cos^2(\theta) = 1$. Therefore, the expression simplifies to 1.

Example 2: Proving an Identity

Prove that $\sec(\theta) - \tan(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$.

Solution:

Starting with the left side of the equation:

[ \begin{align*} \sec(\theta) - \tan(\theta) &= \frac{1}{\cos(\theta)} - \frac{\sin(\theta)}{\cos(\theta)} \ &= \frac{1 - \sin(\theta)}{\cos(\theta)} \ &= \frac{\cos^2(\theta)}{\cos(\theta)\sin(\theta)} \ &= \frac{\cos(\theta)}{\sin(\theta)} \end{align*} ]

Thus, we have shown that $\sec(\theta) - \tan(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$.

Example 3: Solving a Trigonometric Equation

Solve for $\theta$ if $\sin(\theta) = \cos(\theta)$.

Solution:

Using the co-function identity, we know that $\sin(\theta) = \cos(90^\circ - \theta)$. Therefore, we can write:

[ \sin(\theta) = \cos(\theta) = \cos(90^\circ - \theta) ]

This implies that $\theta = 90^\circ - \theta$. Solving for $\theta$ gives us $\theta = 45^\circ$.

Summary Table

Identity Type Identity Example
Pythagorean $\sin^2(\theta) + \cos^2(\theta) = 1$ $\sin^2(30^\circ) + \cos^2(30^\circ) = 1$
Reciprocal $\sin(\theta) = \frac{1}{\csc(\theta)}$ $\sin(45^\circ) = \frac{1}{\csc(45^\circ)}$
Quotient $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ $\tan(60^\circ) = \frac{\sin(60^\circ)}{\cos(60^\circ)}$
Co-Function $\sin(\theta) = \cos(90^\circ - \theta)$ $\sin(30^\circ) = \cos(60^\circ$
Even-Odd $\sin(-\theta) = -\sin(\theta)$ $\sin(-45^\circ) = -\sin(45^\circ)$

Understanding and applying these trigonometric identities is crucial for solving problems in trigonometry, calculus, and other areas of mathematics.