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Chapter 4: Problem 34

Use a calculator to evaluate the trigonometric functions for the indicated angle values. Round your answers to four decimal places. $$\cos 21.9^{\circ}$$

Short Answer

Expert verified
The value of \( cos 21.9^{\circ} \) is approximately 0.9281.

Step by step solution

01

Set Calculator to Degree Mode

Ensure your calculator is in degree mode, as we are dealing with an angle in degrees. This is typically done by pressing the 'Mode' button and selecting 'Degree' instead of 'Radian'.
02

Input the Angle Value

Enter the angle value, 21.9, into your calculator. Make sure to input it correctly to avoid errors.
03

Calculate the Cosine

Use the cosine function by pressing the 'cos' button followed by the angle you've entered. The calculator should display the cosine of 21.9 degrees.
04

Round the Result

Take the result shown on your calculator and round it to four decimal places to obtain the final answer.

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Key Concepts

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

Angle Measurement
Understanding angle measurement is crucial when working with trigonometric functions. Angles are typically measured in degrees or radians, depending on the context. In the problem at hand, the angle is given in degrees, which is one of the most common units used in mathematics.
  • A full circle is 360 degrees.
  • One degree equals 1/360 of a full rotation.
For trigonometric functions like cosine, it is important to be aware of the angle's measurement unit to ensure accurate calculations. When dealing with degrees, visualize the rotation around a circle and how it relates to the angle given. This can help in predicting and understanding the behavior of trigonometric functions.
Degree Mode
When working on problems involving angles measured in degrees, it is essential to set your calculator to degree mode. Calculators can operate in either degree or radian mode, and using the wrong mode can result in incorrect answers.
To set your calculator to degree mode:
  • Generally, press a button labeled 'Mode' or 'Angle'.
  • Select 'Degree' from the options displayed, rather than 'Radian'.
Ensuring your calculator鈥檚 correct mode is a small but vital step. It aligns the calculator's internal settings with the angle measurement given, leading to accurate results while evaluating trigonometric functions.
Cosine Calculation
Calculating the cosine of an angle involves using the cosine function, a fundamental part of trigonometry that relates the ratio of the adjacent side to the hypotenuse in a right triangle. For this exercise:
  • Enter the angle (21.9 degrees) into your calculator.
  • Press the 'cos' button to obtain the cosine value.
This output represents the cosine of the angle 21.9 degrees when measured from the positive x-axis. Cosine calculations are commonly used in solving real-world problems like determining the lengths of sides in a triangle or modeling cycles and waves. Once you have the cosine value, round it to four decimal places. This rounding is often necessary in homework or test situations where increased precision is crucial for demonstrating understanding.

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Most popular questions from this chapter

In calculus we work with real numbers; thus, the measure of an angle must be in radians. An object is rotating at \(600^{\circ}\) per second, find the central angle \(\theta,\) in radians, when \(t=3 \mathrm{sec}\).

Find the angular speed (radians/second) associated with rotating a central angle \(\theta\) in time \(t\). $$\theta=\frac{3 \pi}{4}, t=\frac{1}{6} \mathrm{sec}$$

When light passes from one substance to another, such as from air to water, its path bends. This is called refraction and is what is seen in eyeglass lenses, camera lenses, and gems. The rule governing the change in the path is called Snell's law, named after a Dutch astronomer: \(n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2},\) where \(n_{1}\) and \(n_{2}\) are the indices of refraction of the different substances and \(\theta_{1}\) and \(\theta_{2}\) are the respective angles that light makes with a line perpendicular to the surface at the boundary between substances. The figure shows the path of light rays going from air to water. Assume that the index of refraction in air is \(1 .\) (GRAPH CANNOT COPY) If light rays hit the water's surface at an angle of \(30^{\circ}\) from the perpendicular and are refracted to an angle of \(22^{\circ}\) from the perpendicular, then what is the refraction index for water? Round the answer to two significant digits.

A car owner decides to upgrade from tires with a diameter of 24.3 inches to tires with a diameter of 26.1 inches. If she doesn't update the onboard computer, how fast will she actually be traveling when the speedometer reads 65 mph? Round to the nearest mph.

Determine whether each statement is true or false. \(\sec ^{2} \theta-1\) can be negative for some value of \(\theta\)
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