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 29

Chapter 1, Problem 29

In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. tan θ = -2/3, sin θ > 0

Verified step by step guidance

Identify the given information: \(\tan \theta = -\frac{2}{3}\) and \(\sin \theta > 0\). This tells us the tangent ratio and the sign of the sine function, which helps determine the quadrant where \(\theta\) lies.

Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Since \(\tan \theta\) is negative and \(\sin \theta\) is positive, \(\theta\) must be in the second quadrant (where sine is positive and cosine is negative).

Use the Pythagorean identity to find \(\sin \theta\) and \(\cos \theta\). Let the opposite side be 2 and adjacent side be 3 (from the tangent ratio), then the hypotenuse \(r = \sqrt{2^2 + 3^2} = \sqrt{13}\). Since \(\theta\) is in the second quadrant, \(\sin \theta = \frac{2}{\sqrt{13}}\) and \(\cos \theta = -\frac{3}{\sqrt{13}}\).

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

Express all values in exact form, rationalizing denominators if necessary, to find the exact values of all six trigonometric functions for \(\theta\).

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Key Concepts

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

Trigonometric Function Relationships

Trigonometric functions such as sine, cosine, and tangent are interrelated through identities and ratios. Knowing one function's value, like tangent, allows calculation of others using definitions, e.g., tan θ = sin θ / cos θ, which helps find sine and cosine values.

Sign of Trigonometric Functions in Quadrants

The signs of sine, cosine, and tangent depend on the quadrant where the angle θ lies. Since sin θ > 0 and tan θ = -2/3, θ must be in the second quadrant, where sine is positive and tangent is negative, guiding correct sign assignment for other functions.

Pythagorean Identity

The Pythagorean identity, sin²θ + cos²θ = 1, is essential for finding unknown trigonometric values. After determining sine or cosine from tangent, this identity helps calculate the remaining function values accurately.

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Related Practice

Textbook Question

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.

cos (𝜋/2)

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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/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.

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cot 𝜋/2

Textbook Question

In Exercises 30–32, find the measure of the side of the right triangle whose length is designated by a lowercase letter. Round answers to the nearest whole number.

Textbook Question

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

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.

tan 𝜋