3.3.1: Resources and Key Concepts — Read

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Resources

Conditional Equation: An equation that is true for only a specific set of variable values and is false for others.
Identity: An equation that holds true for all permissible values of the variables involved.
Equivalent Expressions: Two algebraic expressions that produce the same value for all allowable inputs. They form the two sides of an identity.

Ratio Identities:
- For any angle \(\theta\) not coterminal with 90° or 270°, \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).
- For any angle \(\theta\) not coterminal with 0° or 180°, \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\).
Reciprocal Identities:
- \(\sin(\theta) = \frac{1}{\csc(\theta)}\) and \(\csc(\theta) = \frac{1}{\sin(\theta)}\)
- \(\cos(\theta) = \frac{1}{\sec(\theta)}\) and \(\sec(\theta) = \frac{1}{\cos(\theta)}\)
- \(\tan(\theta) = \frac{1}{\cot(\theta)}\) and \(\cot(\theta) = \frac{1}{\tan(\theta)}\)
The Pythagorean Identities: For any angle \(\theta\) where the functions are defined:
- \(\cos^2(\theta) + \sin^2(\theta) = 1\)
- \(1 + \tan^2(\theta) = \sec^2(\theta)\)
- \(\cot^2(\theta) + 1 = \csc^2(\theta)\)
Alternate Forms of the Pythagorean Identity:
- \(\sin^2(\theta) = 1 - \cos^2(\theta) \implies \sin(\theta) = \pm\sqrt{1 - \cos^2(\theta)}\)
- \(\cos^2(\theta) = 1 - \sin^2(\theta) \implies \cos(\theta) = \pm\sqrt{1 - \sin^2(\theta)}\)

Using Examples as Proofs: To prove an equation is an identity, you must use algebraic methods. Showing that an equation works for a few numerical examples is not sufficient, as it does not prove it is true for all possible values.
Incorrectly Taking Square Roots: Taking the square root of both sides of an equation like \(\cos^2(\theta) + \sin^2(\theta) = 1\) does not result in \(\cos(\theta) + \sin(\theta) = 1\). The square root operation does not distribute over addition.
Confusing Reciprocals with Inverses: The cosecant, secant, and cotangent functions are the multiplicative reciprocals of the sine, cosine, and tangent functions, respectively. They are not functional inverses. The term "inverse" has a specific meaning for functions that will be discussed later.

This page titled 3.3.1: Resources and Key Concepts was last modified on Tue, 08 Jul 2025 17:43:52 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.