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Magazine Chapter 10: Problem 24
Solve the equation, giving the exact solutions which lie in \([0,2 \pi)\). $$ \cos (2 x)=2-5 \cos (x) $$
Short Answer
Expert verified
The solutions are \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\).
Step by step solution
01
Use a Double Angle Identity
The equation involves \(\cos(2x)\) which can be expressed in terms of \(\cos(x)\) using a double angle identity. Recall that \(\cos(2x) = 2\cos^2(x) - 1\). Substitute this into the equation:\[2\cos^2(x) - 1 = 2 - 5\cos(x)\]
02
Rearrange the Equation
Bring all terms to one side to form a quadratic equation in terms of \(\cos(x)\).\[2\cos^2(x) + 5\cos(x) - 3 = 0\]
03
Solve the Quadratic Equation
To solve the quadratic equation \(2\cos^2(x) + 5\cos(x) - 3 = 0\), use the quadratic formula \(\cos(x) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = 5\), and \(c = -3\).Calculate the discriminant:\[b^2 - 4ac = 5^2 - 4(2)(-3) = 25 + 24 = 49\]Now solve for \(\cos(x)\):\[\cos(x) = \frac{-5 \pm \sqrt{49}}{4}\]Calculate the solutions:\[\cos(x) = \frac{-5 + 7}{4} = \frac{1}{2}\] \[\cos(x) = \frac{-5 - 7}{4} = -3\]
04
Consider Valid Solutions
The value \(\cos(x) = -3\) is not valid since the cosine function ranges between \(-1\) and \(1\). Therefore, we only consider \(\cos(x) = \frac{1}{2}\).
05
Solve \(\cos(x) = \frac{1}{2}\)
Find the angles \(x\) where \(\cos(x) = \frac{1}{2}\) that lie within \([0, 2\pi)\). These angles are:\[x = \frac{\pi}{3}, \frac{5\pi}{3}\]
06
Final Solution
The exact solutions for \(x\) in the interval \([0, 2\pi)\) are \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\).
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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 a critical concept in trigonometry, often denoted as \( \cos(x) \). It relates to the angle \( x \) of a right triangle and is defined as the adjacent side divided by the hypotenuse. Its values range from \(-1\) to \(1\). In the unit circle, the cosine of an angle is the \(x\)-coordinate of the point where the terminal side of the angle intersects the circle. This function is periodic with a period of \(2\pi\), meaning every \(2\pi\) units, the values repeat.
- The cosine waveform begins at 1 when \(x = 0\), moves down to -1 by \(\pi\), and returns to 1 at \(2\pi\).
- It is an even function, satisfying \(\cos(-x) = \cos(x)\).
Double Angle Identities
Double angle identities are formulas used to express trigonometric functions of double angles, such as \(2x\), in terms of functions of a single angle \(x\). For cosine, this identity is \(\cos(2x) = 2\cos^2(x) - 1\). These identities are essential in simplifying expressions and solving trigonometric equations.
- The identity \(\cos(2x) = 2\cos^2(x) - 1\) allows us to rewrite the equation \(\cos(2x) = 2 - 5\cos(x)\) in terms of \(\cos(x)\).
- Substituting the double angle identity simplifies the problem to a quadratic equation, which can be easier to solve.
Quadratic Equations
A quadratic equation is a second-degree polynomial equation that typically takes the form \(ax^2 + bx + c = 0\). In this context, the equation is \(2\cos^2(x) + 5\cos(x) - 3 = 0\), where the variable is \(\cos(x)\). Quadratic equations are solved using the quadratic formula:\[\cos(x) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
- Here, \(a = 2\), \(b = 5\), and \(c = -3\).
- Calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots.
Exact Solutions
Exact solutions refer to finding precise numerical values, without approximations, for variables that satisfy the equation. In solving trigonometric equations, like \(\cos(x) = \frac{1}{2}\), exact solutions are particularly valued.
- The equation \(\cos(x) = \frac{1}{2}\) is solved by identifying standard angles \(x\) where cosine has this value within the interval \([0, 2\pi)\).
- For cosine, these angles are \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\).
