/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 \(\cos \theta=\frac{5}{12}\)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 14

\(\cos \theta=\frac{5}{12}\)

Short Answer

Expert verified
The angle \( \theta \) is equal to \( \cos^{-1}(\frac{5}{12}) \).

Step by step solution

01

Understand the Problem

The problem is asking for the angle \( \theta \) where \(\cos \theta = \frac{5}{12}\). So, we are looking for an angle \( \theta \) whose cosine is \( \frac{5}{12} \).
02

Use Inverse Cosine

We use the inverse cosine function, also called arccos or cos^-1, to find the angle when the value of cosine is given. So we can find the \( \theta \) as \( \theta = \cos^{-1}(\frac{5}{12}) \).
03

Get the Answer

Then calculate \( \theta = \cos^{-1}(\frac{5}{12}) \) and find the angle. Bear in mind that the result can be in radian, so if the requirement is to provide the result in degrees, please convert it accordingly.

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Key Concepts

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

Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is especially concerned with right-angled triangles, where one of the angles is always 90 degrees. The basic trigonometric functions are sine, cosine, and tangent, which are used to relate the angles of a triangle to the ratios of its sides.

These functions are essential in many areas of mathematics and science, including physics, engineering, and astronomy. They help to solve problems involving right triangles and can be used to analyze periodic functions.
Cosine of Angle
The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. For an angle 胃, it is represented as \( \cos(\theta) \).

The cosine function has a range of values from -1 to 1 and is an even function, meaning it is symmetrical around the y-axis. It is crucial for determining the horizontal component of a vector, calculating forces in physics, and in the analysis of waves and oscillations.
Arccos
Arccos, also known as the inverse cosine function, is denoted as \( \cos^{-1} \) or \( \text{arccos} \). It is used to determine the angle when the value of the cosine is known.

Given a value 'x' within the range of -1 to 1, \( \theta = \text{arccos}(x) \) will give us the angle in radians or degrees for which the cosine equals 'x'. It is important to note that the function will return an angle within the range of 0 to 蟺 (or 0 to 180掳) since the cosine function is positive in the first and fourth quadrants and this is where its inverse is typically defined.
Angle Measurement
Angle measurement is the process of determining the size of an angle in units, either degrees or radians. One full rotation around a point is equivalent to 360 degrees or 2蟺 radians.

Degrees are commonly used in everyday applications such as navigation and construction, while radians are more prevalent in higher mathematics and sciences because of their natural connection to the properties of circles. Being able to convert between these two units is important for solving trigonometry problems effectively.

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Most popular questions from this chapter

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