Trigonometry
Study of triangles and trigonometric functions including sine, cosine, and tangent.
Trigonometric Identities
What Are Trigonometric Identities?
Trigonometric identities are equations involving trig functions that are always true for any angle.
Fundamental Identities
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Pythagorean Identity:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
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Quotient Identities:
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
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Reciprocal Identities:
\[ \csc(\theta) = \frac{1}{\sin(\theta)}, \quad \sec(\theta) = \frac{1}{\cos(\theta)}, \quad \cot(\theta) = \frac{1}{\tan(\theta)} \]
Why Use Identities?
They help simplify complex problems and prove relationships between different trigonometric functions.
Key Formula
\[\sin^2(\theta) + \cos^2(\theta) = 1\]
Examples
Using the Pythagorean identity to find \(\sin(\theta)\) if you know \(\cos(\theta)\).
Proving that \(\tan^2(\theta)+1=\sec^2(\theta)\) using identities.
In a Nutshell
Identities help you manipulate and simplify trigonometric expressions.