/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 Use a calculator to find the fol... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 44

Use a calculator to find the following. $$ \tan 171^{\circ} 40^{\prime} $$

Short Answer

Expert verified
\( \tan 171^{\circ} 40^{\prime} \approx -2.747 \)

Step by step solution

01

Convert Minutes to Decimal Degrees

The given angle is in degrees and minutes: \( 171^{\circ} 40^{\prime} \). Minutes need to be converted to decimal degrees for calculator use. There are 60 minutes in a degree, so the conversion is \( 40^{\prime} = \frac{40}{60} = 0.6667^{\circ} \). Therefore, the angle becomes \( 171.6667^{\circ} \).
02

Use the Calculator to Find the Tangent

With the angle now in decimal form, input it into your calculator as \( \tan 171.6667^{\circ} \). Ensure your calculator is set to degree mode before computing. The result is approximately \( -2.747 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decimal Degrees Conversion
When dealing with trigonometric functions, angles must often be converted into a convenient form for calculation. The decimal degrees conversion process is an essential step. Given angles often come in degrees and minutes format, such as the angle in this problem: \( 171^{\circ} 40^{\prime} \).
To perform this conversion, understand that each degree is made up of 60 minutes. Therefore, to convert minutes to a portion of a degree, divide the number of minutes by 60.
For this exercise:
  • There are 40 minutes.
  • Convert this by calculating \( \frac{40}{60} \), which equals \( 0.6667^{\circ} \).
The angle in decimal degrees is then \( 171^{\circ} + 0.6667^{\circ} = 171.6667^{\circ} \).
This step is crucial for the next calculations, especially when using a calculator.
Degree Mode on Calculator
Before performing any trigonometric calculations on a calculator, it's crucial to ensure the calculator is in the correct mode. Calculators generally offer two modes for angle measurements: degree mode and radian mode. For this problem, you'll need the degree mode.
Why is this important?
  • The tangent function, like other trigonometric functions, varies based on the unit of the angle.
  • If the calculator is in radian mode instead of degree mode, the results will not represent the intended decimal degree calculation, leading to errors.
To set your calculator to degree mode:
  • Look for a button labeled "MODE" or a similar setting menu.
  • Select "DEGREE" from the options provided.
  • Confirm that the display indicates the calculator is now using degrees.
Ensuring the calculator is in the right mode is a simple step that prevents mistakes and ensures accurate results.
Using Calculator for Trigonometric Functions
Calculators are powerful tools for solving trigonometric equations like finding the tangent of an angle. Once your calculator is in degree mode, you can proceed to calculate the tangent of the angle.
Here's how to use your calculator for trigonometric functions:
  • Enter the converted angle: type 171.6667.
  • Press the TAN button to find the tangent of the angle.
  • The calculator will display the result immediately.
For this particular problem:
  • Input the angle \( 171.6667^{\circ} \).
  • The calculator should show a result close to \( -2.747 \).
Understanding how to use your calculator effectively ensures you can quickly and accurately compute trigonometric functions for any angle.

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Most popular questions from this chapter

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