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Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblem 1.2.72
Chapter 1, Problem 1.2.72

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

Verified step by step guidance
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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Related Practice
Textbook Question

In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. sin(-240°)

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Textbook Question

In Exercises 55–58, use a calculator to find the value of the acute angle θ to the nearest degree. sin θ = 0.2974

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Textbook Question

In Exercises 59–62, use a calculator to find the value of the acute angle θ in radians, rounded to three decimal places. cos θ = 0.4112

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Textbook Question

In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. -150°

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Textbook Question

In Exercises 55–58, use a calculator to find the value of the acute angle θ to the nearest degree. tan θ = 4.6252

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Textbook Question

In Exercises 71–74, find the length of the arc on a circle of radius r intercepted by a central angle θ. Express arc length in terms of 𝜋. Then round your answer to two decimal places. Radius, r: 8 feet Central Angle, θ: θ = 225°