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 30

Chapter 1, Problem 30

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.

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Identify the sides of the right triangle relative to the given angle of 48° at vertex P. The side adjacent to the angle is PR = 700 m, the side opposite the angle is QR, and the hypotenuse is PQ.

Use the trigonometric ratios to relate the sides to the angle. For example, to find the hypotenuse PQ, use the cosine function: \(\cos(48^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{700}{PQ}\).

Rearrange the equation to solve for the hypotenuse PQ: \(PQ = \frac{700}{\cos(48^\circ)}\).

To find the opposite side QR, use the sine function: \(\sin(48^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{QR}{PQ}\).

Rearrange the sine equation to solve for QR: \(QR = PQ \times \sin(48^\circ)\). Substitute the expression for PQ from step 3 to find QR in terms of known values.

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Key Concepts

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

Right Triangle Properties

A right triangle has one angle of 90 degrees, and the side opposite this angle is the hypotenuse, the longest side. The other two sides are called legs. Understanding the relationship between these sides and angles is essential for solving problems involving right triangles.

Trigonometric Ratios (Sine, Cosine, Tangent)

Trigonometric ratios relate the angles of a right triangle to the ratios of its sides. Sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. These ratios allow calculation of unknown side lengths or angles when one side and one angle are known.

Using Given Angle and Side to Find Unknown Sides

Given an angle and one side length in a right triangle, you can use trigonometric ratios to find the other sides. For example, with angle 48° and adjacent side 700 m, cosine can find the hypotenuse, and tangent can find the opposite side. Rounding to the nearest whole number is often required.

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

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.

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cot 𝜋/2

Textbook Question

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

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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. 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