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Chapter 5: Problem 32

Evaluate each trigonometric function without the use of a calculator. $$\cos (\arccos (0.8))$$

Short Answer

Expert verified
From above analysis, the solution to \(\cos(\arccos(0.8))\) is 0.8.

Step by step solution

01

Understanding of Inverse Trigonometric Functions

First of all, we recognize that \(\arccos(0.8))\) is an inverse trigonometric function and its output gives an angle such that the cosine of this angle returns the original number - in this case, 0.8. Therefore, any real number within the domain of cosine, i.e., [0, \(\pi\)], is a possible output of \(\arccos(x)\).
02

Applying the Inverse Trigonometric Function

In this step, we apply the inverse trigonometric function \(\arccos(0.8))\). This will give us an angle. Particular to the definition, this angle, let's denote it as \(A\), will satisfy \(\cos(A) = 0.8\). So, we can say that \(A = \arccos(0.8)\).
03

Applying the Cosine function

Now that we have our angle \(A\), we do \(\cos(A)\). However, we've already defined \(A\) such that \(\cos(A) = 0.8\). Therefore, \(\cos(\arccos(0.8) = 0.8\).

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Key Concepts

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

Inverse Trigonometric Functions
Inverse trigonometric functions are essential to unlocking angles when given a trigonometric value. These functions essentially reverse the roles of trigonometric functions. For instance, while the cosine function takes an angle and gives us a ratio, the inverse cosine, or arccosine, takes a ratio and provides an angle.
Inverse trigonometric functions include:
  • Arccosine (\( \arccos \))
  • Arcsine (\( \arcsin \))
  • Arctangent (\( \arctan \))
Each function provides a specific angle that correlates with a given trigonometric ratio. The primary role of these functions is to determine angles in different geometric situations. This is especially useful in various applications, such as when solving triangle problems and analyzing wave functions.
Cosine
The cosine function is commonly used to determine the x-coordinate of a point on a unit circle at a given angle. It is a trigonometric function and part of the foundation for understanding trigonometry. The cosine of an angle is the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
Some features of the cosine function:
  • Periodicity: Repeats its values every \( 2\pi \) radians.
  • Range: Cosine values are between -1 and 1, inclusive.
  • Domain: Typically, all real numbers. But when associated with inverse functions, it is often restricted.
The cosine function's formula is written as \( \cos(\theta) \). This function is crucial when finding the lengths of sides in a triangle or the x-component of vectors. Its precise nature and simplicity make it an indispensable tool in both pure and applied mathematics.
Arccosine
Arccosine is the inverse of the cosine function. Its primary purpose is to find the angle that corresponds to a given cosine value. In other words, if you know the cosine of an angle, \( \arccos \) helps determine that angle.
Characteristics of the arccosine function include:
  • Domain: The range of values for input is from -1 to 1.
  • Range: Output angles are restricted between 0 and \( \pi \) radians, capturing all possible scenarios of the cosine values crossing from -1 to 1.
  • Utilization: Commonly used in locating angles in computations where the cosine is known.
Understanding arccosine is key to unraveling problems where reverse calculation from trigonometric values to angles is needed. This is particularly beneficial in geometry and certain branches of physics, where angle calculations are crucial to problem-solving.

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Most popular questions from this chapter

In this set of exercises, you will use degree and radian measure to study real-world problems. In the 1800 s, women often carried pleated fans. One of the fans on display at the Smithsonian is 7 inches long and, when fully open, sweeps out an angle of \(80^{\circ}\) How long is the trim, to the nearest tenth of an inch, on the curved edge of the fan?

Find the sine and cosine of the angle \(z\) in \([0,2 \pi),\) in standard position, cohose terminal side intersects the unit circle at the giecn point. $$(0.8,-0.6)$$

Graph the given pair of functions in the same window. Graph at least two cycles of each function, and describe the similarities and differences between the graphs. $$f(x)=\sec \left(\frac{\pi}{2} x\right) ; f(x)=\sec (2 \pi x)$$

The first ferris wheel was 250 feet in diameter. It was invented by John Ferris in \(1893 .\) Assuming it made one revolution every 30 seconds, what was the angular speed of a passenger (assume the passenger is on the edge of the wheel) in degrees per minute? What was the passenger's linear speed in feet per minute?

Use the negative-angle identities to compute the exact value of each of the given trigonometric functions. $$\csc \left(-\frac{5 \pi}{4}\right)$$
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