/*! 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 23 Use a calculator to find the val... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 23

Use a calculator to find the values of the following trigonometric functions. \(\cos 12^{\circ}\)

Short Answer

Expert verified
0.9781

Step by step solution

01

- Understand the Problem

Given the trigonometric function \(\forall \cos 12^{\circ}\). The task is to find its value.
02

- Set the Calculator to the Correct Mode

Ensure that the calculator is set to degree mode since the angle given is in degrees. This can typically be done by pressing the 'Mode' button and selecting 'Degree'.
03

- Enter the Function into the Calculator

Enter the following sequence of buttons on your calculator to find the cosine of 12 degrees: 'cos', followed by '12', and then press 'Enter' or '='.
04

- Record the Answer

The calculator should display the value. \(\forall \cos 12^{\circ} \approx 0.9781\)

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Key Concepts

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

cosine function
The cosine function is one of the fundamental trigonometric functions, usually abbreviated as 'cos'. It's used to measure the adjacent side of a right triangle in relation to the hypotenuse. Cosine values range from -1 to 1. When you see something like \(\cos 12^{\circ}\), it means you are finding the adjacent side's length divided by the hypotenuse for a 12-degree angle.
An important thing to remember is that the cosine function is periodic and symmetrical. This means it repeats its values in predictable patterns and looks the same on both sides of certain angles. This is especially useful in both math and physics problems.
using a calculator
Using a calculator for trigonometric functions like the cosine can be very helpful and fast. Here鈥檚 a simple guide to follow:
  • First, find a calculator that has trigonometric functions. Most scientific calculators will have this.
  • Ensure your calculator can switch between different modes, like degrees and radians. This is often done using the 'Mode' button.
  • To find the cosine of an angle, press the 'cos' button, then type the angle value, and finally hit 'Enter' or '=' to get the result.
Always double-check the mode your calculator is in, because the wrong mode will give incorrect results, especially for trigonometric calculations.
degree mode
Degree mode is essential when dealing with angles measured in degrees. Calculators can be set to different modes: degree or radian. Since \(12^{\circ}\) is given in degrees, your calculator must be set to degree mode to get the correct value.
Here鈥檚 how you typically set it:
  • Press the 'Mode' button on your calculator.
  • Select the 'Degree' option by navigating through the menu.
Switching to the correct mode is crucial because if the calculator were in radian mode, the value for \(\cos 12^{\circ}\) would be calculated incorrectly. This is a common mistake, so always check before performing your calculations.

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Most popular questions from this chapter

Amplitude and Mid-Line. Consider the function \(g(t)=a \sin b t+k\), where \(a\) and \(b\) are positive. a) Show that the maximum and minimum of \(g(t)\) are \(k+a\) and \(k-a\), respectively. b) Show that the mid-line is the line \(y=k\). c) Show that the amplitude of \(g(t)\) is \(a\).

Oscillations with Trends. Graph the following functions on the interval \(0 \leq t \leq 10\). $$ f(t)=t^{2} / 10+3+4 \sin 5 t $$

Sound Waves. The pitch of a sound wave is measured by its frequency. Humans can hear sounds in the range from 20 to \(20,000 \mathrm{~Hz}\), while dogs can hear sounds as high as \(40,000 \mathrm{~Hz}\). The loudness of the sound is determined by the amplitude. \({ }^{22}\). Blood Prcssur: During a period of controlled breathing, the systolic blood pressure \(p\) of a volunteer averaged \(143 \mathrm{mmHg}\) with an amplitude of \(5.3 \mathrm{mmHg}\) and a frequency of \(0.172 \mathrm{~Hz}\). Assuming that the blood pressure was highest when \(t=0\), find a model \(p(t)=a \cos b t+k\) for blood pressure as a function of time.

Sound Waves. The pitch of a sound wave is measured by its frequency. Humans can hear sounds in the range from 20 to \(20,000 \mathrm{~Hz}\), while dogs can hear sounds as high as \(40,000 \mathrm{~Hz}\). The loudness of the sound is determined by the amplitude. \({ }^{22}\). The note A above middle \(C\) on a piano generates a sound modeled by the function \(g(t)=4 \sin (880 \pi t)\), where \(t\) is in seconds. Find the frequency of \(\mathrm{A}\) above middle \(\mathrm{C}\).

On a 5 -mi stretch of highway, the road decreases in elevation at an angle of \(4^{\circ} .\) How much lower is a car after traveling on this part of the highway? (Remember that there are 5280 feet in a mile.)
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