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 28–29, find a cofunction with the same value as the given expression. sin 70°

Verified step by step guidance

Recall the cofunction identity for sine and cosine: \(\sin(\theta) = \cos(90^\circ - \theta)\).

Identify the angle in the given expression: here, \(\theta = 70^\circ\).

Apply the cofunction identity by substituting \(\theta\) with \(70^\circ\): \(\sin 70^\circ = \cos(90^\circ - 70^\circ)\).

Simplify the expression inside the cosine: \(90^\circ - 70^\circ = 20^\circ\).

Write the final cofunction expression: \(\sin 70^\circ = \cos 20^\circ\).

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Key Concepts

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

Cofunction Identity

Cofunction identities relate trigonometric functions of complementary angles, meaning angles that add up to 90°. For sine and cosine, sin(θ) = cos(90° - θ). This identity allows us to find a cofunction with the same value by subtracting the given angle from 90°.

Complementary Angles

Complementary angles are two angles whose measures add up to 90°. Understanding this concept is essential because cofunction identities depend on the relationship between complementary angles, enabling the conversion between sine and cosine values.

Evaluating Trigonometric Functions at Specific Angles

Evaluating trigonometric functions at specific angles, such as 70°, involves understanding angle measures and their corresponding function values. This skill helps in applying cofunction identities correctly to find equivalent expressions.

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Textbook Question

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

Textbook Question

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.

cos 9𝜋/2

Textbook Question

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

cos (𝜋/2)

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Textbook Question

In Exercises 21–28, convert each angle in radians to degrees. -3𝜋

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Textbook Question

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

Textbook Question

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