/*! 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 120 Solve the equation. $$6 \cos x... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 120

Solve the equation. $$6 \cos x-2=1$$

Short Answer

Expert verified
The solutions for the equation \(6 \cos x - 2 = 1\) are \(x = 60掳, 300掳\) or \(x = \pi/3, 5\pi/3\) in radians.

Step by step solution

01

Rearrange the equation

It is important to first isolate the term involving the cos(x). To do that, the equation \(6 \cos x - 2 = 1\) can be rearranged to \(6 \cos x = 3\). This is done by adding 2 to both sides of the equation.
02

Further isolate to \(\cos x\)

The equations should now be divided by 6 on both sides to fully isolate the \(\cos x\) on one side. This will result in the equation: \(\cos x = 3/6\), which simplifies to \(\cos x = 0.5\).
03

Find the value of x

The final step is to use the inverse cosine function to solve for \(x\). This gives us: \(x = \arccos(0.5)\). The arccosine of 0.5 returns two solutions 60掳 and 300掳 in degrees or \(\pi/3\) and \(5\pi/3\) in radians if the domain is [0, 2pi].

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Key Concepts

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

Cosine Function
The cosine function is an essential trigonometric function. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Understanding cosine is key in solving various trigonometric equations, like the one in our exercise. In a unit circle, the cosine of an angle gives the x-coordinate of the point where the terminal side of the angle intersects the circle.
  • The cosine function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) radians.
  • Its range is between -1 and 1, inclusive of both.
Despite its simplicity, the cosine function is incredibly powerful in mathematics and physics, particularly for modeling periodic phenomena like sound waves and light.
Inverse Trigonometric Functions
Inverse trigonometric functions are vital for finding angles given a trigonometric ratio. Specifically, the inverse cosine function, denoted as \(\arccos\), helps us determine the angle whose cosine value is known.When solving an equation like \(\cos x = 0.5\), the task is to find out the value of \(x\) such that the cosine of \(x\) equals 0.5.
  • \(\arccos(0.5)\) leads to an angle measurement of both \(\pi/3\) and \(5\pi/3\) within the interval [0, 2\pi].
  • These solutions reflect the fact that cosine is positive in both the first and fourth quadrants of the unit circle.
  • Using inverse trigonometric functions, you can determine multiple solutions due to the periodic nature of trig functions.
Inverse trigonometric functions open a doorway to converting difficult trigonometric expressions into solvable problems, making them indispensable for certain types of mathematical problem-solving.
Radian Measure
Radian measure is another way to express angle magnitudes and is commonly used in trigonometry. Unlike degrees, which divide a circle into 360 sections, radians relate the angle measurement directly to the arc length of a circle.One radian is defined as the angle created when the radius is wrapped along the arc of the circle.
  • There are approximately \(6.2832\), or \(2\pi\), radians in a complete circle.
  • An angle of \(\pi/3\) radians is equivalent to 60 degrees, which matches our solution from the exercise.
  • Radian measure simplifies mathematical formulas, particularly in calculus, making it more efficient in various computations.
Using radians is particularly beneficial when working within the realms of higher mathematics and physics, where precision and mathematical elegance are paramount.

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