Problem 50 Find the exact solutions of the ... [FREE SOLUTION]

Chapter 6: Problem 50

Find the exact solutions of the given equations, in radians, that lie in the interval $[0,2 \pi)$. $$\cos 2 x-5 \cos x-2=0$$

Short Answer

Expert verified

The exact solutions of the equation that satisfy $\cos 2 x-5 \cos x-2=0$ and lie in the interval $[0,2 \pi)$ are $x = \dfrac{2\pi}{3}$ or $x = \dfrac{4\pi}{3}$.

Step by step solution

Substitute double-angle identity for cosine

Replace $\cos{2x}$ in the equation with $2\cos^2{x} - 1$, obtaining $2\cos^2{x} - 5\cos{x} - 2 -1 = 0$. Consequently, the equation simplifies to $2\cos^2{x} - 5\cos{x} -3 = 0$. This step is important because it transformed the original equation into a quadratic equation, which makes it simpler to solve.

Solve the quadratic equation

The quadratic equation $2\cos^2{x} - 5\cos{x} -3 = 0$ can be solved by either factoring, completing the square or using the quadratic formula. By applying the quadratic formula, we get $\cos x = \dfrac{5\pm \sqrt{((5)^2 - 4(2)(-3))}}{2*2}$, which simplifies to $\cos x = \dfrac{5\pm \sqrt{49}}{4}$ leading ultimately to $\cos{x} = 2$ or $\cos{x} = -\dfrac{1}{2}$.

Solve for $x$

Once you have $\cos{x} = 2$ and $\cos{x} = -\dfrac{1}{2}$, you can solve for $x$. When $\cos{x} = 2$, there is no solution as the range of cosine function is $[-1,1]$. When $\cos{x} = -\dfrac{1}{2}$, $x = \dfrac{2\pi}{3}$ or $x = \dfrac{4\pi}{3}$ are solutions, considering the definition of cosine function in a unit circle and the given interval $[0,2\pi)$.

Verify the solution

The solutions $x = \dfrac{2\pi}{3}$ and $x = \dfrac{4\pi}{3}$ obtained should satisfy the given equation when substituted back into the equation to confirm them as valid solutions within the specified interval.

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Key Concepts

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

Double-Angle Identity

To tackle trigonometric equations, one helpful tool is the double-angle identity. This identity relates trigonometric functions of double angles to single angles, which can simplify complex expressions. For the cosine function, the double-angle identity is expressed as:

$ \cos{2x} = 2\cos^2{x} - 1 $
or equivalently, $ \cos{2x} = \cos^2{x} - \sin^2{x} $

In our problem, we used this identity to replace $ \cos{2x} $ with $ 2\cos^2{x} - 1 $. This step simplifies the equation, transforming it into a quadratic form, which is far easier to solve. Recognizing when and how to apply these identities is crucial for simplifying and solving trigonometric equations effectively.

Quadratic Equation

Once the trigonometric equation is converted using the double-angle identity, it takes on the form of a quadratic equation. Quadratic equations are polynomial equations of the form $ ax^2 + bx + c = 0 $. Solving them involves finding the values of $ x $ that satisfy the equation.

You can solve quadratic equations by:
- Factoring
- Completing the square
- Applying the quadratic formula

For this particular exercise, we applied the quadratic formula, $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, to find $ \cos{x} = 2 $ or $ \cos{x} = -\frac{1}{2} $. This was a crucial turning point in understanding how cosine values can be derived from the quadratic form within trigonometric equations.

Unit Circle

The unit circle is a fundamental concept in trigonometry crucial for understanding the behavior of trigonometric functions. It is a circle with a radius of one, centered at the origin of a coordinate plane. Trigonometric functions, including sine and cosine, can be associated with angles on the unit circle.

For any angle $ x $, $ \cos{x} $ represents the x-coordinate on the unit circle.
This means that valid values for the cosine function range from $-1$ to $1$.

In our problem, the solution $ \cos{x} = 2 $ was discarded because it lies outside the valid range for cosine. Understanding the unit circle helps determine valid solutions and further aids in finding angles that correspond to given trigonometric values, like $ \cos{x} = -\frac{1}{2} $.

Cosine Function

The cosine function is one of the primary trigonometric functions closely related to the sides of a right triangle and the unit circle. In a coordinate plane, $ \cos{x} $ determines the horizontal distance from the origin.

The range of $ \cos{x} $ is $[-1, 1]$, indicating the maximum and minimum values it can take.
Functions like the double-angle identities can manipulate expressions involving cosine.

In this exercise, solving $ \cos{x} = 2 $ was invalid due to the cosine's range. However, for $ \cos{x} = -\frac{1}{2} $, we found solutions: $ x = \frac{2\pi}{3} $ and $ x = \frac{4\pi}{3} $ within the interval $[0,2\pi)$. Mastery in managing these cosine values is essential for effective problem-solving in trigonometry.

91影视

Find the exact solutions of the given equations, in radians, that lie in the interval \([0,2 \pi)\). $$\cos 2 x-5 \cos x-2=0$$

Short Answer

Step by step solution

Substitute double-angle identity for cosine

Solve the quadratic equation

Solve for \(x\)

Verify the solution

Key Concepts

Double-Angle Identity

Quadratic Equation

Unit Circle

Cosine Function

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