Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problem 2.3.48

Chapter 3, Problem 2.3.48

Use a calculator to evaluate each expression. 2 sin 25°13' cos 25°13' - sin 50°26'

Verified step by step guidance

First, recognize that the angles are given in degrees and minutes. Convert the angle 25°13' into decimal degrees if needed, or use the angle directly in your calculator by inputting degrees and minutes appropriately.

Recall the double-angle identity for sine: \(\sin(2\theta) = 2 \sin \theta \cos \theta\). Notice that the expression \(2 \sin 25°13' \cos 25°13'\) matches the left side of this identity with \(\theta = 25°13'\).

Rewrite the expression \(2 \sin 25°13' \cos 25°13'\) as \(\sin(2 \times 25°13')\), which simplifies to \(\sin 50°26'\).

Substitute this back into the original expression to get \(\sin 50°26' - \sin 50°26'\).

Since the two terms are the same, their difference is zero. This shows the expression simplifies to zero without needing further calculation.

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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 Values

Sine and cosine are fundamental trigonometric functions that relate angles to ratios of sides in a right triangle. Understanding how to evaluate these functions for given angles, including those expressed in degrees and minutes, is essential for solving the expression.

Angle Conversion and Notation

Angles given in degrees and minutes (e.g., 25°13') must be accurately interpreted or converted to decimal degrees for calculator input. One minute equals 1/60 of a degree, so converting ensures precise evaluation of trigonometric functions.

Trigonometric Identities

The expression involves terms like 2 sin A cos A and sin 2A, which are connected by the double-angle identity: sin 2A = 2 sin A cos A. Recognizing and applying this identity simplifies the expression and aids in verifying the result.

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Fundamental Trigonometric Identities

Related Practice

Textbook Question

CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.

Column I: 1.

cot⁻¹ 30

Column II:

A. 88.09084757°

B. 63.25631605°

C. 1.909152433°

D. 17.45760312°

E. 0.2867453858

F. 1.962610506

G. 14.47751219°

H. 1.015426612

I. 1.051462224

J. 0.9925461516

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Textbook Question

Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. Suppose that an observer for a radar station is located at the origin of a coordinate system. Find the bearing of an airplane located at each point. Express the bearing using both methods. (0, -2)

Textbook Question

Use a calculator to evaluate each expression. cos 75°29' cos 14°31' - sin 75°29' sin 14°31'

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Textbook Question

Use a calculator to evaluate each expression. sin 35° cos 55° + cos 35° sin 55°

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Textbook Question

Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. 2 cos 38°22' = cos 76°44'

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Textbook Question

Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1.

sin 38° 42'

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