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Chapter 7: Problem 19

$$\text {Solve each equation for exact solutions over the interval }[0,2 \pi) .$$ $$\cos ^{2} x+2 \cos x+1=0$$

Short Answer

Expert verified
The exact solution over the interval \([0, 2\pi)\) is \( x = \pi\)

Step by step solution

01

Recognize the quadratic form

Rewrite the equation \(\cos^2 x + 2\cos x + 1 = 0\) as a quadratic equation in \(\cos x\)
02

Simplify the quadratic equation

Notice that the given equation can be factored as \((\cos x + 1)^2 = 0\)
03

Solve for \(\cos x\)

Solve the equation \((\cos x + 1)^2 = 0\). The solution is \(\cos x = -1\)
04

Find exact solutions in the given interval

Determine the values of \(x\) within the interval \([0, 2\pi)\) that satisfy \(\cos x = -1\). The only solution is \( x = \pi\)

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Key Concepts

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

Quadratic Equation
A quadratic equation is an equation of the form \[ax^2 + bx + c = 0\]. In our exercise, we start by recognizing that \(\text{cos}^2 x + 2 \text{cos} x + 1 = 0\) is quadratic in terms of \(\text{cos} x\).
This means that \( \text{cos} x \) can be considered as the variable, just like \(x\) in a typical quadratic equation.
Identifying the quadratic form helps us apply methods that we use for solving quadratic equations. These steps will include factoring, finding roots, and verifying solutions.
Keep in mind:
  • The standard form of a quadratic equation is \(ax^2 + bx + c = 0\)
  • Comparison to \(\text{cos} ^{2} x+2 \text{cos} x+1=0\) shows \(a = 1, b = 2, \text{ and } c = 1\)
Cosine Function
The cosine function, \(\text{cos}(x)\), is one of the fundamental trigonometric functions.
It describes the horizontal coordinate (or x-coordinate) of a point on the unit circle corresponding to an angle \(x\). In our equation, \( \text{cos} x\) is treated as a variable that we solve for.
Understanding the Cosine Function:
  • It has a range of \([-1, 1]\)
  • Its values repeat every \(2\text{蟺}\) radians, making it a periodic function
Application to the problem:
In our equation \(\text{cos}x = -1\), we need to find the angle \(x\) for which \(\text{cos} x\) equals \(-1\) in the interval \([0, 2\text{蟺})\).
Factoring
Factoring is the process of breaking down an equation into simpler components that can be solved more easily.
In the given problem, we factor \(\text{cos}^2 x + 2 \text{cos} x + 1 = 0\) by noticing it can be rewritten as \((\text{cos} x + 1)^2 = 0\).
This is a significant step because it simplifies solving the equation.
Here鈥檚 how:
  • If \((\text{cos} x + 1)^2 = 0\), then \(\text{cos} x + 1\) must be equal to zero
  • Thus, \(\text{cos} x = -1\)
Finding Solutions:
The next step is to determine the values of \(x\) within the interval \([0, 2\text{蟺})\) for which the cosine equals \(-1\). The exact solution is \(x = \text{蟺}\).

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Most popular questions from this chapter

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$1-\sin x=\cos 2 x$$

Verify that each equation is an identity. $$\sin 2 x=2 \sin x \cos x$$

Verify that each equation is an identity. $$\sin 4 x=4 \sin x \cos x \cos 2 x$$

Explain why attempting to find \(\sin ^{-1} 1.003\) on your calculator will result in an error message.

Amperage, Wattage, and Voltage Amperage is a measure of the amount of electricity that is moving through a circuit, whereas voltage is a measure of the force pushing the electricity. The wattage \(W\) consumed by an electrical device can be determined by calculating the product of the amperage \(I\) and voltage \(V .\) (Source: Wilcox, G. and C. Hesselberth, Electricity for Engineering Technology, Allyn \& Bacon.)(a) A household circuit has voltage $$ V=163 \sin 120 \pi t $$ when an incandescent light bulb is turned on with amperage $$ I=1.23 \sin 120 \pi t $$ Graph the wattage \(W=V I\) consumed by the light bulb in the window \([0,0.05]\) by $$ [-50,300] $$ (b) Determine the maximum and minimum wattages used by the light bulb. (c) Use identities to determine values for \(a, c,\) and \(\omega\) so that \(W=a \cos (\omega t)+c\) (d) Check your answer by graphing both expressions for \(W\) on the same coordinate axes. (e) Use the graph to estimate the average wattage used by the light. For how many watts (to the nearest integer) do you think this incandescent light bulb is rated?
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