Ch. 1 - Angles and the Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 1 - Angles and the Trigonometric Functions

Problem 33

Chapter 1, Problem 33

In Exercises 31–38, find a cofunction with the same value as the given expression. csc 25°

Verified step by step guidance

Recall the definition of a cofunction: for an angle \( \theta \), the cofunction identity states that \( \sin(90^\circ - \theta) = \cos \theta \) and similarly for other trigonometric functions, such as \( \csc(\theta) = \sec(90^\circ - \theta) \).

Identify the given function: \( \csc 25^\circ \) is the cosecant of 25 degrees, which is the reciprocal of sine, i.e., \( \csc \theta = \frac{1}{\sin \theta} \).

Use the cofunction identity for cosecant: \( \csc \theta = \sec(90^\circ - \theta) \). This means \( \csc 25^\circ = \sec(90^\circ - 25^\circ) \).

Calculate the complementary angle inside the cofunction: \( 90^\circ - 25^\circ = 65^\circ \).

Write the cofunction with the same value as the original expression: \( \csc 25^\circ = \sec 65^\circ \).

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Key Concepts

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

Cosecant Function (csc)

The cosecant function is the reciprocal of the sine function, defined as csc θ = 1/sin θ. It is used to find the ratio of the hypotenuse to the opposite side in a right triangle. Understanding csc is essential to relate it to other trigonometric functions.

Cofunction Identity

Cofunction identities relate trigonometric functions of complementary angles, such as sin(90° - θ) = cos θ. These identities help find equivalent expressions by switching between pairs like sine and cosine or tangent and cotangent for angles summing to 90°.

Complementary Angles

Complementary angles are two angles whose measures add up to 90°. In trigonometry, many function values at an angle θ correspond to cofunctions at 90° - θ, enabling simplification or transformation of expressions using cofunction identities.

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Intro to Complementary & Supplementary Angles

Related Practice

Textbook Question

Find a cofunction with the same value as the given expression.

csc 35°

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Textbook Question

In Exercises 29–34, convert each angle in degrees to radians. Round to two decimal places. -50°

Textbook Question

In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c. tan(-t) - tan t

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Textbook Question

Find a cofunction with the same value as the given expression.

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Textbook Question

In Exercises 31–38, find a cofunction with the same value as the given expression. tan 𝜋 9

Textbook Question

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. sec θ = -3, tan θ > 0

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