๐ Pythagorean Identities
Pythagorean Identities are derived from the Pythagorean theorem ($a^2 + b^2 = c^2$) applied to the unit circle. They relate the squares of trigonometric functions.
- ๐ The most fundamental Pythagorean Identity is: $\sin^2(\theta) + \cos^2(\theta) = 1$
- ๐ก Dividing the fundamental identity by $\cos^2(\theta)$ yields: $1 + \tan^2(\theta) = \sec^2(\theta)$
- ๐ Dividing the fundamental identity by $\sin^2(\theta)$ gives us: $\cot^2(\theta) + 1 = \csc^2(\theta)$
๐ Reciprocal Identities
Reciprocal Identities define the relationships between a trigonometric function and its reciprocal. They are essentially 'flip' relationships.
- ๐ Sine and Cosecant: $\csc(\theta) = \frac{1}{\sin(\theta)}$
- ๐ก Cosine and Secant: $\sec(\theta) = \frac{1}{\cos(\theta)}$
- ๐ Tangent and Cotangent: $\cot(\theta) = \frac{1}{\tan(\theta)}$
๐ Pythagorean Identities vs. Reciprocal Identities: A Comparison
| Feature |
Pythagorean Identities |
Reciprocal Identities |
| Definition |
Relate squares of trigonometric functions based on the Pythagorean theorem. |
Define a trigonometric function as the reciprocal of another. |
| Form |
Involve squares and sums/differences: $\sin^2(\theta) + \cos^2(\theta) = 1$ |
Involve reciprocals: $\csc(\theta) = \frac{1}{\sin(\theta)}$ |
| Use Case |
Simplifying expressions, proving other identities, solving equations. |
Simplifying expressions, converting between functions. |
| Key Functions |
Sine, Cosine, Tangent, Secant, Cosecant, Cotangent (all are related within the three identities) |
Pairs of functions that are reciprocals of each other (e.g., sine and cosecant). |
๐ Key Takeaways
- ๐ Pythagorean Identities come from the Pythagorean Theorem, while Reciprocal Identities come from the definition of reciprocal functions.
- ๐ก Pythagorean Identities involve squares of trigonometric functions, while Reciprocal Identities do not.
- ๐ Both types of identities are essential tools for simplifying trigonometric expressions and solving equations.