Problem 4 - Precalculus

Question

All of the Pythagorean identities are related. Describe how to manipulate the equations to get from \(\sin ^{2} t+\cos ^{2} t=1\) to the other forms.

Step-by-Step Solution

Answer

Use division by \( \cos^2 t \) for \( \tan^2 t + 1 = \sec^2 t \) and \( \sin^2 t \) for \( 1 + \cot^2 t = \csc^2 t \).

Step 1: Start with the Primary Identity

The primary Pythagorean identity is given by the equation \( \sin^2 t + \cos^2 t = 1 \). This identity is fundamental and describes the relationship between the sine and cosine functions.

Step 2: Derive the Tangent-Secant Identity

Divide each term in the primary identity by \( \cos^2 t \) to derive the tangent-secant identity:\[ \frac{\sin^2 t}{\cos^2 t} + \frac{\cos^2 t}{\cos^2 t} = \frac{1}{\cos^2 t} \].This simplifies to \( \tan^2 t + 1 = \sec^2 t \).

Step 3: Derive the Cotangent-Cosecant Identity

Similarly, divide each term in the primary identity by \( \sin^2 t \) to derive the cotangent-cosecant identity:\[ \frac{\sin^2 t}{\sin^2 t} + \frac{\cos^2 t}{\sin^2 t} = \frac{1}{\sin^2 t} \].This simplifies to \( 1 + \cot^2 t = \csc^2 t \).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential formulas in trigonometry that show the fundamental relationships between trigonometric functions. These identities help simplify complex expressions and solve trigonometric equations. One of the most crucial identities you will encounter is the Pythagorean identity, given by:

• \( \sin^2 t + \cos^2 t = 1 \)

This identity expresses the inherent relationship between the sine and cosine functions of any angle \(t\). By understanding this identity, you can derive other key identities used in various mathematical computations and proofs.
The power of trigonometric identities lies in their ability to transform expressions, allowing for easier manipulation in solving trigonometric problems. Knowing how to manipulate and transform these identities can greatly aid in simplifying expressions and solving equations involving trigonometric functions.

Tangent-Secant Identity

The tangent-secant identity is another key trigonometric identity derived from the primary Pythagorean identity. To derive this identity, you start with:

• \( \sin^2 t + \cos^2 t = 1 \)

Then, divide every term by \( \cos^2 t \):

• \( \frac{\sin^2 t}{\cos^2 t} + \frac{\cos^2 t}{\cos^2 t} = \frac{1}{\cos^2 t} \)

This simplifies to:

• \( \tan^2 t + 1 = \sec^2 t \)

Here, \( \tan t = \frac{\sin t}{\cos t} \) and \( \sec t = \frac{1}{\cos t} \). By transforming the identity in this manner, you've expressed a relationship between tangent and secant.
This is particularly useful in calculus and higher-level mathematics, where you often deal with trigonometric ratios and need to convert between different functions. Understanding how to derive and apply this identity can simplify the process of solving trigonometric equations.

Cotangent-Cosecant Identity

The cotangent-cosecant identity is another valuable identity derived from the primary Pythagorean identity. To find this identity, you start with:

• \( \sin^2 t + \cos^2 t = 1 \)

Simply divide each term by \( \sin^2 t \):

• \( \frac{\sin^2 t}{\sin^2 t} + \frac{\cos^2 t}{\sin^2 t} = \frac{1}{\sin^2 t} \)

This leads to:

• \( 1 + \cot^2 t = \csc^2 t \)

Here, \( \cot t = \frac{\cos t}{\sin t} \) and \( \csc t = \frac{1}{\sin t} \). This identity provides a unique relationship among cotangent and cosecant.
This form of identity is especially useful in analytical trigonometry and problem-solving situations where you need to change expressions or find solutions involving cotangent and cosecant. Understanding these connections helps in exploring deeper trigonometric relations.