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Chapter 8: Problem 42

Find the measure of each angle to the nearest tenth of a degree. $$\cos D=0.1212$$

Short Answer

Expert verified
The measure of angle \( D \) is approximately 82.0 degrees.

Step by step solution

01

Use Inverse Cosine

To find the angle when given the cosine value, we use the inverse cosine (arc cosine) function. The arc cosine, denoted as \( \cos^{-1} \), allows us to determine the angle whose cosine value is known.
02

Calculate the Angle

Use a calculator to find \( D = \cos^{-1}(0.1212) \). Ensure your calculator is set to degree mode before computing. Input the value to get an angle measurement.
03

Round the Angle

Once you have the angle measurement from the calculation, round it to the nearest tenth of a degree as required by the exercise.

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Key Concepts

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

Understanding Inverse Cosine
When you have a cosine value and want to determine the corresponding angle, inverse cosine is your go-to tool. The inverse cosine function is often represented as \( \cos^{-1} \). It helps in finding the angle when the value of cosine is known. Essentially, you are asking, "What angle has this cosine value?"

Here鈥檚 how it works:
  • The range of typical cosine values is from -1 to 1.
  • The result of the inverse cosine is always an angle.
  • This function is crucial in right triangle trigonometry to find missing angles.
It鈥檚 important to ensure your calculator or software is set to the correct mode (degrees or radians) depending on what you need. In most real-world problems like the one in the exercise, you鈥檒l want degree mode.
Angle Measurement Essentials
Angles are measured in degrees or radians, but degrees are more common in most practical situations.

Understanding how to measure angles is crucial:
  • A full circle is 360 degrees.
  • Each degree can be divided into 60 minutes, and each minute into 60 seconds.
  • However, for simplicity, decimal degrees are often used.
When encountering a problem where you need to find an angle from a trigonometric function (like cosine), always double-check your calculator settings to be sure they match the units you鈥檙e working in. In our exercise, we're dealing with degrees, so multiply any results if needed when you want to switch between degrees and radians (using \( \pi \)).
Rounding to the Nearest Tenth
Rounding is vital to present a number in a simpler, more comprehendible form without losing significant accuracy. Here's how you can round a number to the nearest tenth.

Consider a number you need to round, look at the digit in the tenths place (the first digit to the right of the decimal point).
  • If the next digit to the right (the hundredths place) is 5 or greater, round up the tenths place digit by one.
  • If it is less than 5, keep the tenths place digit as it is.
For example, suppose you calculate the angle as 82.345 degrees. By examining the number, the hundredths place digit is 4, which is less than 5. Therefore, you would not increase the tenths place, and it would round to 82.3 degrees, aligning with the exercise requirement of rounding to the nearest tenth.

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Most popular questions from this chapter

Determine whether each set of numbers can be the measures of the sides of a right triangle. Then state whether they form a Pythagorean triple. $$8,15,17$$

Find each measure using the given measures of \(\triangle X Y Z .\) Round angle measures to the nearest degree and side measures to the nearest tenth. If \(y=17, z=14,\) and \(m \angle Y=92,\) find \(m \angle Z\)

use the following information. Mr. Martinez is an architect who designs houses so that the windows receive minimum Sun in the summer and maximum Sun in the winter. For Columbus, Ohio, the angle of elevation of the Sun at noon on the longest day is \(73.5^{\circ}\) and on the shortest day is \(26.5^{\circ} .\) Suppose a house is designed with a south-facing window that is 6 feet tall. The top of the window is to be installed 1 foot below the overhang. How long should the architect make the overhang so that the window gets no direct sunlight at noon on the longest day?

Solve each \(\triangle P Q R\) described below. Round angle measures to the nearest degree and side measures to the nearest tenth. $$m \angle P=33, m \angle R=58, q=22$$

Determine whether each set of measures can be the sides of a right triangle. Then state whether they form a Pythagorean triple. $$5,12,13$$
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