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 2.3.18

Chapter 3, Problem 2.3.18

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. tan(-80° 06')

Verified step by step guidance

First, convert the angle from degrees and minutes to a decimal degree format. Since there are 60 minutes in a degree, convert 6 minutes to degrees by dividing 6 by 60: \(6' = \frac{6}{60} = 0.1^\circ\).

Add this decimal to the degrees part of the angle: \(-80^\circ 06' = -80^\circ - 0.1^\circ = -80.1^\circ\).

Recall that the tangent function is periodic with period \(180^\circ\), so you can use the identity \(\tan(\theta) = \tan(\theta + 180^\circ k)\) for any integer \(k\) to simplify the angle if needed. However, in this case, the angle is already within a common range for calculator input.

Use a calculator set to degree mode to find \(\tan(-80.1^\circ)\). Make sure the calculator is in degree mode, not radians.

Finally, round the result to six decimal places as requested.

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Key Concepts

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

Angle Measurement and Conversion

Angles can be expressed in degrees, minutes, and seconds, where 1 degree equals 60 minutes. To use a calculator, angles in degrees and minutes must be converted to decimal degrees by dividing the minutes by 60 and adding to the degrees. For example, -80° 06' converts to -80.1°.

Tangent Function and Its Properties

The tangent function relates an angle in a right triangle to the ratio of the opposite side over the adjacent side. It is periodic with period 180°, and its values can be positive or negative depending on the angle's quadrant. Understanding the sign and behavior of tangent helps interpret results correctly.

Using Calculators for Trigonometric Approximations

Calculators typically require angles in decimal degrees or radians to compute trigonometric values. After converting the angle, use the calculator's tangent function to find the approximate value. Rounding the result to six decimal places ensures precision as requested.

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