Ch. 5 - Trigonometric Identities

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problema 5.74

Capitolo 6, Problema 5.74

Verify that each equation is an identity.
sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² θ)

Guida verificata passo dopo passo

Step 1: Start by simplifying the left-hand side (LHS) of the equation: \( \frac{\sin \theta}{1 - \cos \theta} - \frac{\sin \theta \cos \theta}{1 + \cos \theta} \).

Step 2: Find a common denominator for the fractions on the LHS: \((1 - \cos \theta)(1 + \cos \theta) = 1 - \cos^2 \theta = \sin^2 \theta\).

Step 3: Rewrite each fraction with the common denominator: \( \frac{\sin \theta (1 + \cos \theta) - \sin \theta \cos \theta (1 - \cos \theta)}{\sin^2 \theta} \).

Step 4: Simplify the numerator: \( \sin \theta + \sin \theta \cos \theta - \sin \theta \cos \theta + \sin \theta \cos^2 \theta = \sin \theta + \sin \theta \cos^2 \theta \).

Step 5: Factor out \( \sin \theta \) from the numerator: \( \frac{\sin \theta (1 + \cos^2 \theta)}{\sin^2 \theta} = \csc \theta (1 + \cos^2 \theta) \), which matches the right-hand side (RHS).

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Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include the Pythagorean identities, reciprocal identities, and co-function identities. Understanding these identities is crucial for simplifying trigonometric expressions and verifying equations as identities.

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Fundamental Trigonometric Identities

Reciprocal Functions

Reciprocal functions in trigonometry relate the sine, cosine, and tangent functions to their respective cosecant, secant, and cotangent functions. For example, csc θ is the reciprocal of sin θ, defined as 1/sin θ. Recognizing these relationships helps in transforming and simplifying expressions, particularly when verifying identities.

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Secant, Cosecant, & Cotangent on the Unit Circle

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. This includes factoring, combining like terms, and applying common denominators. Mastery of these techniques is essential for verifying trigonometric identities, as it allows one to transform one side of the equation to match the other.

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Algebraic Operations on Vectors

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