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 27

Chapter 1, Problem 27

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. cos θ = 8/17, 270° < θ < 360°

Verified step by step guidance

Identify the quadrant where the angle \( \theta \) lies. Since \( 270^\circ < \theta < 360^\circ \), \( \theta \) is in the fourth quadrant, where cosine is positive and sine is negative.

Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Use this to find \( \sin \theta \) by substituting \( \cos \theta = \frac{8}{17} \).

Calculate \( \sin \theta = -\sqrt{1 - \cos^2 \theta} = -\sqrt{1 - \left(\frac{8}{17}\right)^2} \) because sine is negative in the fourth quadrant.

Find \( \tan \theta \) using the definition \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Substitute the values of \( \sin \theta \) and \( \cos \theta \) found in previous steps.

Determine the remaining trigonometric functions using their relationships: \( \csc \theta = \frac{1}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), and \( \cot \theta = \frac{1}{\tan \theta} \).

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Key Concepts

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

Trigonometric Functions and Their Relationships

Trigonometric functions include sine, cosine, tangent, and their reciprocals cosecant, secant, and cotangent. Knowing one function value, such as cosine, allows you to find others using identities and relationships, like sin²θ + cos²θ = 1.

Quadrant Sign Rules

The sign of trigonometric functions depends on the quadrant of the angle. Since 270° < θ < 360° (fourth quadrant), cosine is positive, sine is negative, and tangent is negative. This helps determine the correct sign of each function value.

Using the Pythagorean Identity to Find Sine

Given cos θ, sine can be found using sin²θ = 1 - cos²θ. After calculating the magnitude, the quadrant determines the sign of sine. This step is essential to find all remaining trigonometric functions accurately.

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Pythagorean Identities

Related Practice

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In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator.sin 37° csc 37°

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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.

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

In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of0, 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.

sin 11𝜋/4

Textbook Question

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

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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.

cos 3𝜋/4

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