Ratios & Identities: Fundamental identities

Ratios & Identities: Fundamental Identities

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The subject is built on the foundation of trigonometric ratios and identities. These ratios and identities are essential for solving various problems in trigonometry, calculus, and other areas of mathematics.

Trigonometric Ratios

Trigonometric ratios are functions of an angle used to relate the angles of a triangle to the lengths of its sides. There are six fundamental trigonometric ratios:

  1. Sine (sin)
  2. Cosine (cos)
  3. Tangent (tan)
  4. Cosecant (csc)
  5. Secant (sec)
  6. Cotangent (cot)

These ratios are defined for a right-angled triangle as follows:

  • Sine (sin): Opposite side / Hypotenuse
  • Cosine (cos): Adjacent side / Hypotenuse
  • Tangent (tan): Opposite side / Adjacent side
  • Cosecant (csc): Hypotenuse / Opposite side
  • Secant (sec): Hypotenuse / Adjacent side
  • Cotangent (cot): Adjacent side / Opposite side

Fundamental Identities

Fundamental identities are equations that hold true for all values of the variable involved, provided the expressions are defined. These identities are used to simplify trigonometric expressions and solve trigonometric equations. The most important fundamental identities are:

  1. Pythagorean Identities
  2. Reciprocal Identities
  3. Quotient Identities
  4. Co-Function Identities
  5. Even-Odd Identities

Pythagorean Identities

The Pythagorean identities are derived from the Pythagorean theorem and relate the squares of the primary trigonometric functions.

  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

Reciprocal identities express the relationship between the primary trigonometric functions and their reciprocals.

  1. $\sin\theta = \frac{1}{\csc\theta}$
  2. $\cos\theta = \frac{1}{\sec\theta}$
  3. $\tan\theta = \frac{1}{\cot\theta}$
  4. $\csc\theta = \frac{1}{\sin\theta}$
  5. $\sec\theta = \frac{1}{\cos\theta}$
  6. $\cot\theta = \frac{1}{\tan\theta}$

Quotient Identities

Quotient identities define 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

Co-function identities show the relationship between the trigonometric functions and their complementary angles.

  1. $\sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta$
  2. $\cos\left(\frac{\pi}{2} - \theta\right) = \sin\theta$
  3. $\tan\left(\frac{\pi}{2} - \theta\right) = \cot\theta$
  4. $\csc\left(\frac{\pi}{2} - \theta\right) = \sec\theta$
  5. $\sec\left(\frac{\pi}{2} - \theta\right) = \csc\theta$
  6. $\cot\left(\frac{\pi}{2} - \theta\right) = \tan\theta$

Even-Odd Identities

Even-odd identities describe the symmetry properties of trigonometric functions.

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

Examples

To illustrate the use of these identities, let's look at a few examples.

Example 1: Simplifying an Expression

Simplify the expression $\sin^2\theta \cdot \sec^2\theta$.

Solution:

Using the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ and the reciprocal identity $\sec\theta = \frac{1}{\cos\theta}$, we can rewrite the expression as:

$$ \sin^2\theta \cdot \sec^2\theta = \sin^2\theta \cdot \left(\frac{1}{\cos\theta}\right)^2 = \sin^2\theta \cdot \frac{1}{\cos^2\theta} $$

Now, using the Pythagorean identity $\sin^2\theta = 1 - \cos^2\theta$, we get:

$$ \sin^2\theta \cdot \frac{1}{\cos^2\theta} = (1 - \cos^2\theta) \cdot \frac{1}{\cos^2\theta} = \frac{1}{\cos^2\theta} - 1 = \sec^2\theta - 1 $$

Example 2: Proving an Identity

Prove that $\tan\theta + \cot\theta = \sec\theta \cdot \csc\theta$.

Solution:

Start by expressing the left side of the equation using quotient identities:

$$ \tan\theta + \cot\theta = \frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} $$

Find a common denominator and combine the terms:

$$ \frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} = \frac{\sin^2\theta + \cos^2\theta}{\sin\theta \cdot \cos\theta} $$

Using the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$, we have:

$$ \frac{\sin^2\theta + \cos^2\theta}{\sin\theta \cdot \cos\theta} = \frac{1}{\sin\theta \cdot \cos\theta} $$

Finally, apply the reciprocal identities:

$$ \frac{1}{\sin\theta \cdot \cos\theta} = \frac{1}{\sin\theta} \cdot \frac{1}{\cos\theta} = \csc\theta \cdot \sec\theta $$

Thus, we have proven that $\tan\theta + \cot\theta = \sec\theta \cdot \csc\theta$.

Understanding and applying these fundamental identities is crucial for solving trigonometric problems and is a key skill for students preparing for exams in mathematics.