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Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblema 1.2.72
Capitolo 1, Problema 1.2.72

If θ is an acute angle and cos θ = 1/3, find csc (𝜋/2 - θ).

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1
Recall the co-function identity for sine and cosine: \(\sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta\).
Since \(\csc x = \frac{1}{\sin x}\), express \(\csc\left(\frac{\pi}{2} - \theta\right)\) as \(\frac{1}{\sin\left(\frac{\pi}{2} - \theta\right)}\).
Substitute the co-function identity into the expression: \(\csc\left(\frac{\pi}{2} - \theta\right) = \frac{1}{\cos\theta}\).
Use the given value \(\cos\theta = \frac{1}{3}\) to rewrite the expression as \(\csc\left(\frac{\pi}{2} - \theta\right) = \frac{1}{\frac{1}{3}}\).
Simplify the fraction to find the expression for \(\csc\left(\frac{\pi}{2} - \theta\right)\) in terms of the given cosine value.

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Complementary Angles in Trigonometry

Complementary angles are two angles whose measures add up to 90° (or π/2 radians). In trigonometry, the sine of an angle equals the cosine of its complement, i.e., sin(π/2 - θ) = cos θ. This relationship helps simplify expressions involving complementary angles.
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Intro to Complementary & Supplementary Angles

Reciprocal Trigonometric Functions

Reciprocal functions are the inverses of the basic trigonometric functions. For example, cosecant (csc) is the reciprocal of sine, defined as csc θ = 1/sin θ. Understanding this helps in converting between sine and cosecant values to solve problems.
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Introduction to Trigonometric Functions

Using Given Values to Find Trigonometric Ratios

Given a trigonometric value like cos θ = 1/3, you can find related ratios using identities. Since sin(π/2 - θ) = cos θ, knowing cos θ allows direct calculation of sin(π/2 - θ), and thus csc(π/2 - θ) by taking the reciprocal. This approach simplifies problem-solving.
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