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 28

Chapter 1, Problem 28

In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator. sin² 𝜋 + cos² 𝜋 10 10

Verified step by step guidance

Recall the Pythagorean identity in trigonometry: \(\sin^2 x + \cos^2 x = 1\) for any angle \(x\).

Identify the angle given in the problem, which is \(\frac{\pi}{10}\).

Apply the identity directly by substituting \(x = \frac{\pi}{10}\) into the formula: \(\sin^2 \left(\frac{\pi}{10}\right) + \cos^2 \left(\frac{\pi}{10}\right)\).

Since the identity holds for all angles, the expression simplifies to 1 without needing to calculate the sine or cosine values individually.

Therefore, the value of \(\sin^2 \frac{\pi}{10} + \cos^2 \frac{\pi}{10}\) is 1.

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Key Concepts

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

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²θ + cos²θ = 1. This fundamental trigonometric identity is derived from the Pythagorean theorem and is essential for simplifying expressions involving sine and cosine squared terms.

Evaluating Trigonometric Functions at Specific Angles

Understanding the values of sine and cosine at key angles, such as π (180 degrees), helps in directly substituting and simplifying expressions. For example, sin(π) = 0 and cos(π) = -1, which are critical for evaluating the given expression without a calculator.

Using Identities to Simplify Expressions

Applying trigonometric identities allows simplification of complex expressions into simpler forms. Instead of calculating sine and cosine values separately, identities like sin²θ + cos²θ = 1 provide a straightforward way to find the value of the expression efficiently.

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If θ is an acute angle and sin θ = (2√7) / 7, use the identity sin²θ + cos²θ = 1 to find cos θ.

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In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 3𝜋, 𝜋, 5𝜋, 3𝜋, 7𝜋, and 2𝜋. 4 2 4 4 2 4 a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function. b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.

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In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator. sec² 23° - tan² 23°

Textbook Question

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

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Textbook Question

In Exercises 29–34, convert each angle in degrees to radians. Round to two decimal places. 18°