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Chapter 5: Problem 23

Use a calculator to find the value of each expression, rounded to four decimal places. $$\sin 27^{\circ}$$

Short Answer

Expert verified
The value of \( \sin 27^{\circ} \) is approximately 0.4540, rounded to four decimal places.

Step by step solution

01

Understand the Expression

We need to find the value of \( \sin 27^{\circ} \), where \( 27^{\circ} \) is an angle in degrees. The sine function gives the ratio of the opposite side to the hypotenuse in a right triangle.
02

Set Up Your Calculator

Ensure that your calculator is set to degree mode since the angle provided is in degrees. This is crucial for getting the correct sine value.
03

Enter the Expression into the Calculator

Input \( \sin 27 \) into your calculator. Be sure to use the sine function button, usually labeled as "SIN" and enter 27 after verifying the calculator is in the correct mode.
04

Calculate and Round

Once you have entered \( \sin 27 \) and pressed the equal button, the calculator should display a result. Round this result to four decimal places as required by the problem.

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Key Concepts

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

Understanding the Sine Function
The sine function is one of the primary trigonometric functions used to relate angles to sides of a right triangle. In any right triangle:
  • The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
  • The sine function's values range from -1 to 1.
For example, if you imagine a right triangle with an angle of 27 degrees, the sine of 27 degrees (\(\sin 27^{\circ}\)) will be the ratio of the opposite side over the hypotenuse. This is integral in fields like physics and engineering.
It's important to remember that sine values are determined by the angle - altering the angle will change the sine value.
Working with Degrees
Angles can be measured in degrees or radians, but in many introductory problems, degrees are common. Here's what you need to know:
  • Degrees divide a circle into 360 equal parts.
  • When you see a degree symbol (掳), it indicates the measurement is in degrees.
For instance, when you see \(27^{\circ}\), it denotes an angle that is 27 degrees from the reference line, such as the horizontal axis in unit circle scenarios. Always ensure your calculations follow the correct unit by aligning your calculator settings appropriately.
Using a Calculator Correctly
For solving trigonometric problems like \(\sin 27^{\circ}\), using a calculator effectively is essential:
  • Make sure your calculator is in degree mode. Devices often default to radian mode, so double-check this setting.
  • Input the sine function using the designated "SIN" button, followed by the angle value.
  • Use the buttons correctly to ensure no notation errors occur, affecting your result.
By following these steps, you avoid common pitfalls and ensure that you compute the sine value accurately. Calculators are powerful tools when used correctly, guiding you to the expected answer efficiently.
Rounding Numbers
When a problem or exercise instructs you to round a result, it's crucial to follow these steps:
  • Identify how many decimal places are needed. In this problem, four decimal places are specified.
  • Look at the fifth decimal place to decide if the fourth should round up.
  • If the fifth place is 5 or more, round up. Otherwise, keep the fourth place as is.
For instance, if your calculator presents a sine value of 0.4540 after calculating \(\sin 27^{\circ}\), this number already meets the four-decimal-place requirement. Precise rounding ensures that your answers remain accurate and adhere to the instructions given.

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Most popular questions from this chapter

The height of an equilateral triangle: \(H=\frac{\sqrt{3}}{2} S\) Given an equilateral triangle with sides of length \(S\), the height of the triangle is given by the formula shown. Once the height is known the area of the triangle can easily be found (also see Exercise 93 ). The Gateway Arch in St. Louis, Missouri, is actually composed of stainless steel sections that are equilateral triangles. At the base of the arch the length of the sides is \(54 \mathrm{ft}\) The smallest cross section at the top of the arch has sides of \(17 \mathrm{ft}\). Find the area of these cross sections.

At carnivals and fairs, the Gravity Drum is a popular ride. People stand along the wall of a circular drum with radius \(12 \mathrm{ft},\) which begins spinning very fast, pinning them against the wall. The drum is then turned on its side by an armature, with the riders screaming and squealing with delight. As the drum is raised to a near-vertical position, it is spinning at a rate of 35 rpm. (a) What is the angular velocity in radians? (b) What is the linear velocity (in miles per hour) of a person on this ride?

Use the formula for area of a circular sector to find the value of the unknown quantity: \(A=\frac{1}{2} r^{2} \theta\). $$A=1080 \mathrm{mi}^{2} ; r=60 \mathrm{mi}$$

Use a calculator to find the acute angle whose corresponding ratio is given. Round to the nearest 10 th of a degree. $$\sin A=0.9063$$

One of the four blades on a ceiling fan has a decal on it and begins at a designated "12 o'clock" position. Turning the switch on and then immediately off, causes the blade to make over three complete, counterclockwise rotations, with the blade stopping at the 8 o'clock position. What angle \(\theta\) did the blade turn through? Name all angles that are coterminal with \(\theta\)
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