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Chapter 8: Problem 8

Use a calculator to find each value. Round to the nearest ten-thousandth. $$\sin 85^{\circ}$$

Short Answer

Expert verified
The value of \( \sin 85^{\circ} \) rounded to the nearest ten-thousandth is approximately 0.9962.

Step by step solution

01

Set Up the Calculation

To begin, we need to determine the value of \( \sin 85^{\circ} \) using a calculator. Make sure your calculator is in degree mode since the angle is given in degrees.
02

Input the Angle

Enter \( 85 \) into the calculator. Look for a button labeled 'SIN' or 'sin' on the calculator and press this button after entering the angle.
03

Read the Calculator Output

Once you press the 'SIN' button, the calculator will display the sine value of \( 85^{\circ} \). Here, you are looking for the decimal output.
04

Round the Result

Take the calculator's output and round it to the nearest ten-thousandth. The ten-thousandth place is the fourth digit after the decimal point.

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Key Concepts

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

Sine Function
The sine function is a fundamental concept in trigonometry. It helps us understand relationships in a right triangle, particularly with angles and side lengths. Think of it as a way to connect angles to ratios of a triangle's sides. When you are dealing with the sine function, you're essentially looking at the ratio of the length of the side opposite the angle to the length of the hypotenuse. This can be expressed mathematically as:
  • For an angle \( \theta \), the sine function is written as \( \sin(\theta) \).
  • In a right triangle, \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \).
In the context of using a calculator, the sine of an angle will give you a decimal value that represents this ratio. You just input the angle, press the 'SIN' button, and that's your sine value.
Rounding Decimals
Rounding decimals is a way to make long numbers easier to work with and understand. It's a way to approximate a number by eliminating less significant digits. When rounding to a specific decimal place, you look at the number immediately to the right of your desired decimal position.
  • If that number is 5 or greater, you round up by adding 1 to your target decimal place.
  • If it is less than 5, you simply leave the target decimal place unchanged.
In the case of rounding to the nearest ten-thousandth:
  • The ten-thousandth place is the fourth digit to the right of the decimal point.
  • For example, to round 0.123456 to the nearest ten-thousandth, you look at the number in the fifth place, which is 5, and then round up to get 0.1235.
Mastering this process is crucial in various math problems to keep calculations neat and precise.
Degree Mode
Degree mode is an essential setting on a calculator when dealing with angles in trigonometry. It's critical to ensure your calculator is in this mode when you're working with degree measures, as opposed to radians.Here's why degree mode is important:
  • Degrees and radians are two common units of angle measurement in trigonometry.
  • When you input angles measured in degrees into your calculator, it must be in degree mode for accurate calculations.
Switching to degree mode is often simple. Check your calculator 鈥� you usually toggle the mode by pressing a button labeled 'MODE', then selecting 'Deg'. Using degree mode correctly ensures that your calculations, like finding \( \sin 85^{\circ} \), are executed accurately without errors related to incorrect angle measurement units.

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Most popular questions from this chapter

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