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 38

Chapter 1, Problem 38

Find a cofunction with the same value as the given expression.
cos (3𝜋/8)

Verified step by step guidance

Recall the cofunction identity for cosine and sine: \(\cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right)\).

Identify the angle \(\theta\) in the given expression, which is \(\frac{3\pi}{8}\).

Substitute \(\theta = \frac{3\pi}{8}\) into the cofunction identity to get \(\cos\left(\frac{3\pi}{8}\right) = \sin\left(\frac{\pi}{2} - \frac{3\pi}{8}\right)\).

Simplify the expression inside the sine function: \(\frac{\pi}{2} - \frac{3\pi}{8} = \frac{4\pi}{8} - \frac{3\pi}{8} = \frac{\pi}{8}\).

Write the final cofunction expression: \(\cos\left(\frac{3\pi}{8}\right) = \sin\left(\frac{\pi}{8}\right)\).

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Key Concepts

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

Cofunction Identities

Cofunction identities relate the trigonometric functions of complementary angles, such as sin(θ) = cos(π/2 - θ). These identities allow us to express one trigonometric function in terms of another by using the complementary angle, which is essential for finding equivalent expressions.

Complementary Angles

Complementary angles are two angles whose measures add up to π/2 radians (90 degrees). Understanding this concept is crucial because cofunction identities depend on the relationship between an angle and its complement to simplify or rewrite trigonometric expressions.

Evaluating Trigonometric Expressions in Radians

Trigonometric functions often use radian measure, where π radians equal 180 degrees. Being comfortable converting and interpreting angles in radians, such as 3π/8, helps in applying identities and finding equivalent expressions accurately.

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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.

sin(-t - 2𝜋) - cos(-t - 4𝜋) - tan(-t - 𝜋)

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Textbook Question

Find the reference angle for each angle.

205°

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Textbook Question

In Exercises 37–38, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ, or state that the function is undefined.

(0, -1)

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Textbook Question

In Exercises 35–40, convert each angle in radians to degrees. Round to two decimal places.

-4.8 radians

Textbook Question

In Exercises 39–48, use a calculator to find the value of the trigonometric function to four decimal places.

sin 38°

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Textbook Question

In Exercises 31–38, find a cofunction with the same value as the given expression. cos 2𝜋 5