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 31

Chapter 1, Problem 31

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. tan θ = 4/3, cos θ < 0

Verified step by step guidance

Identify the given information: \(\tan \theta = \frac{4}{3}\) and \(\cos \theta < 0\). Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

Determine the quadrant of \(\theta\) based on the signs of \(\tan \theta\) and \(\cos \theta\). Since \(\tan \theta\) is positive and \(\cos \theta\) is negative, \(\theta\) lies in the second quadrant.

Use the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{4}{3}\) to express \(\sin \theta\) and \(\cos \theta\) in terms of a common variable. Let \(\cos \theta = x\), then \(\sin \theta = \frac{4}{3} x\).

Apply the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) by substituting \(\sin \theta = \frac{4}{3} x\) and \(\cos \theta = x\), then solve for \(x\).

Once \(\cos \theta\) and \(\sin \theta\) are found, use these values to find the remaining trigonometric functions: \(\sec \theta = \frac{1}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), 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.

Understanding Trigonometric Ratios

Trigonometric ratios relate the angles of a right triangle to the ratios of its sides. Given tan θ = 4/3, this means the opposite side over adjacent side is 4/3. Knowing one ratio allows calculation of the other trigonometric functions using definitions and relationships.

Sign of Trigonometric Functions in Quadrants

The sign of trigonometric functions depends on the quadrant where the angle lies. Since cos θ < 0, θ is in either the second or third quadrant. This information helps determine the correct signs of sine, cosine, and other functions when calculating their exact values.

Using the Pythagorean Identity

The Pythagorean identity, sin²θ + cos²θ = 1, allows finding missing trigonometric values when one function is known. After determining the sides from tan θ, this identity helps calculate sine and cosine values, ensuring the results satisfy the fundamental trigonometric relationship.

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Related Practice

Textbook Question

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

0, 𝜋/4, 𝜋/2, 3𝜋/4, 𝜋, 5𝜋/4, 3𝜋/2, 7𝜋/4, and 2𝜋.

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.

cot 15𝜋/2

Textbook Question

In Exercises 31–38, find a cofunction with the same value as the given expression. sin 7°

Textbook Question

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

0, 𝜋/4, 𝜋/2, 3𝜋/4, 𝜋, 5𝜋/4, 3𝜋/2, 7𝜋/4, and 2𝜋.

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.

<IMAGE>

cot 𝜋/2

Textbook Question

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

Textbook Question

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

sin 19°

Textbook Question

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

0, 𝜋/4, 𝜋/2, 3𝜋/4, 𝜋, 5𝜋/4, 3𝜋/2, 7𝜋/4, and 2𝜋.

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.

<Image>

sin 47𝜋/4