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Chapter 5: Problem 42

Use a calculator to find the value of the trigonometric function to four decimal places. $$ \tan 52.6^{\circ} $$

Short Answer

Expert verified
The value of \( \tan(52.6^{\circ}) \) to four decimal places is approximately 1.3038

Step by step solution

01

Setup the Calculator

Before starting, check that your calculator is set to use degrees (not radians), because the given angle is in degrees. This is usually done in the settings of the calculator.
02

Input the angle

Input the given angle into the calculator. This is typically done by typing '52.6' (without the quotes) using the numeric keypad.
03

Calculate Tangent

Use the tangent function on the calculator, usually denoted as 'tan', to calculate the value of the tangent of the given angle. Most calculators require you to press a button marked 'tan' after you've entered the angle.
04

Get the result

The calculator will display the value of the tangent of the given angle. Record the result to four decimal places.

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Key Concepts

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

Tangent
Trigonometric functions are essential in understanding relationships within triangles. One such function is the tangent, often abbreviated as "tan." The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side.
If you imagine a right triangle, and you're looking at one of its angles, the formula can be written as: \[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]This function is commonly used in various fields, including physics, engineering, and computer science. By using the tangent, we can find unknown lengths or angles in a triangle provided other information is known.
  • Use to solve right triangles.
  • A basic concept in trigonometry.
  • Simply the ratio of two sides.
Angle in Degrees
When dealing with trigonometric functions, it is crucial to know the measurement type of angles being used鈥攑articularly whether the angle is in degrees, as opposed to radians. For most everyday applications, such as adjusting satellite dishes or measuring ground slope, degrees are more familiar and convenient.
Degrees divide a circle into 360 even parts, where each angle represents one degree. In trigonometry, calculations can drastically change if the calculator is set incorrectly to radians instead of degrees.
  • 360 degrees equal a full circle.
  • One degree is 1/360 of a circle.
  • Always confirm the unit of angle measurement on your device.
Calculator Usage
Using a calculator properly is key to finding accurate values of trigonometric functions. Before any calculation, you need to ensure your calculator is set to the correct mode, especially when using degrees. Most calculators have a mode setting button that lets you switch between degrees (DEG) and radians (RAD).
Once set in degrees, you can enter the angle value directly. For example, you would type '52.6' on the keypad and then use the 'tan' feature to get the tangent of the angle. Many calculators perform this operation following the principle 'angle first, function second.'
  • Always check mode before performing calculations.
  • Enter the angle, followed by the function.
  • Ensures accuracy in mathematical computations.
Decimal Precision
Precision is vital when reporting or using calculated values. In mathematics and science, results are often required to a certain number of decimal places to ensure accuracy and consistency.
In the case of calculating tangent values, providing four decimal places ensures that the result is precise enough for practical applications. Rounding rules should be carefully followed to avoid errors:
  • Round only the final result, not intermediate calculations.
  • Use four decimal places for clarity and precision in results.
  • Helps maintain standardization in scientific reporting.

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Most popular questions from this chapter

Your neighborhood movie theater has a 25 -foot-high screen located 8 feet above your eye level. If you sit too close to the screen, your viewing angle is too small, resulting in a distorted picture. By contrast, if you sit too far back, the image is quite small, diminishing the movie's visual impact. If you sit \(x\) feet back from the screen, your viewing angle, \(\theta,\) is given by $$\theta=\tan ^{-1} \frac{33}{x}-\tan ^{-1} \frac{8}{x}$$ (GRAPH CANNOT COPY) Find the viewing angle, in radians, at distances of 5 feet, 10 feet, 15 feet, 20 feet, and 25 feet.

Let \(f(x)=\left\\{\begin{array}{ll}{x^{2}+2 x-1} & {\text { if } x \geq 2} \\\ {3 x+1} & {\text { if } x<2}\end{array}\right.\) Find \(f(5)-f(-5)\)

A water wheel has a radius of 12 feet. The wheel is rotating at 20 revolutions per minute. Find the linear speed, in feet per minute, of the water.

Exercises \(127-129\) will help you prepare for the material covered in the next section. In each exercise, let \(\theta\) be an acute angle in a right triangle, as shown in the figure. These exercises require the use of the Pythagorean Theorem. If \(a=1\) and \(b=1,\) find the ratio of the length of the side opposite \(\theta\) to the length of the hypotenuse. Simplify the ratio by rationalizing the denominator.

For years, mathematicians were challenged by the following problem: What is the area of a region under a curve between two values of \(x ?\) The problem was solved in the seventeenth century with the development of integral calculus. Using calculus, the area of the region under \(y=\frac{1}{x^{2}+1},\) above the \(x\) -axis, and between \(x=a\) and \(x=b\) is \(\tan ^{-1} b-\tan ^{-1} a\). Use this result, shown in the figure, to find the area of the region under \(y=\frac{1}{x^{2}+1}\) above the \(x\) -axis, and between the values of a and b given in Exercises \(97-98\). (GRAPH CANNOT COPY) \(a=0\) and \(b=2\)
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