Problem 3
Question
Find all solutions to the following equations: (a) \(\sec ^{2} x=2\). (b) \(\cos ^{2} x=0.2 \cos x\). Why can't you cancel the \(\cos x\) from both sides of the equation? (c) \(\sin ^{2} x=3 \cos x+1\).
Step-by-Step Solution
Verified Answer
Solutions for (a): \(x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\). Solutions for (b): \(x = \frac{\pi}{2}, \frac{3\pi}{2}, \cos^{-1}(0.2), 2\pi - \cos^{-1}(0.2)\). Solutions for (c): \(x = \pi, 2\pi\). The \(\cos x\) can't be canceled from both sides because this would also eliminate the solutions where \(\cos x = 0\).
1Step 1: Solution to (a)
First, note that the secant can be rewritten as 1/cosine. So, \(\sec ^{2} x=2\) can be rewritten as \((1/\cos ^{2} x) = 2\). Solving for \(\cos ^{2} x\), we have \(\cos ^{2} x = 1/2\), and taking the square root of both sides, we get \(\cos x = \pm \sqrt{1/2}\), or \(\cos x = \pm 1/\sqrt{2}\). These solutions correspond to \(x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\) in one circle.
2Step 2: Solution to (b) & Explanation of the mentioned issue
The equation \(\cos ^{2} x = 0.2 \cos x\) implies that either \(\cos x = 0\) or \(\cos x = 0.2\). From this we find solutions \(x = \frac{\pi}{2}, \frac{3\pi}{2}\) from \(\cos x = 0\), and \(x = \cos^{-1}(0.2), 2\pi - \cos^{-1}(0.2)\) from \(\cos x = 0.2\). The reason we can't cancel the \(\cos x\) from both sides is because this would also eliminate the solution \(x = \frac{\pi}{2}, \frac{3\pi}{2}\) where \(\cos x = 0\).
3Step 3: Solution to (c)
We have \(\sin ^{2} x = 3 \cos x + 1\), and using the identity \(\sin ^{2} x = 1 - \cos ^{2} x\), we get \(1 - \cos ^{2} x = 3 \cos x + 1\). Rearranging, we get the quadratic equation \(\cos ^{2} x + 3\cos x = 0\). Factoring, we find \(\cos x (\cos x + 3) = 0\), which has solutions \(x = \pi, 2\pi\).
Key Concepts
Secant FunctionCosine FunctionSine FunctionTrigonometric Identities
Secant Function
The secant function is one of the fundamental trigonometric functions. Defined as the reciprocal of the cosine function, it expresses the relationship between the hypotenuse and the adjacent side of a right-angled triangle. Mathematically, the secant of an angle x is represented as \( \sec x = \frac{1}{\cos x} \).
When we encounter the equation \( \sec^2 x = 2 \), as seen in the textbook exercise, we are looking for angles where the secant function assumes the value of \(\sqrt{2}\). To solve this, we transform the secant function to its corresponding cosine function, which is more commonly used and understood. This step of converting secant to cosine simplifies the equation to one involving a basic trigonometric function, making it easier to solve and understand.
When we encounter the equation \( \sec^2 x = 2 \), as seen in the textbook exercise, we are looking for angles where the secant function assumes the value of \(\sqrt{2}\). To solve this, we transform the secant function to its corresponding cosine function, which is more commonly used and understood. This step of converting secant to cosine simplifies the equation to one involving a basic trigonometric function, making it easier to solve and understand.
Cosine Function
The cosine function maps the angle of a right triangle to the ratio of the length of the adjacent side over the hypotenuse. The cosine function's values range from -1 to 1, and it is an even function, meaning \( \cos(x) = \cos(-x) \).
Understanding the cosine function is crucial when solving exercises like \( \cos^2 x = 0.2 \cos x \) because the cosine squared term suggests a second-degree equation. One key point is knowing not to cancel out the cosine terms on both sides, as it would disregard any solutions where \( \cos x = 0 \). Such insights show the importance of understanding not just the functions themselves, but also the implications of the algebraic operations on these functions.
Understanding the cosine function is crucial when solving exercises like \( \cos^2 x = 0.2 \cos x \) because the cosine squared term suggests a second-degree equation. One key point is knowing not to cancel out the cosine terms on both sides, as it would disregard any solutions where \( \cos x = 0 \). Such insights show the importance of understanding not just the functions themselves, but also the implications of the algebraic operations on these functions.
Sine Function
The sine function is another core trigonometric function, representing the ratio of the length of the opposite side to the hypotenuse in a right triangle. For an angle x, it is denoted as \( \sin x \). The values for the sine function also lie between -1 and 1.
In our textbook problem, \( \sin^2 x = 3 \cos x + 1 \), we leverage the relationship between the sine and cosine functions to find a solution. By applying the fundamental trigonometric identity \( \sin^2 x + \cos^2 x = 1 \), we can manipulate the equation and solve it in terms of a single trigonometric function—either sine or cosine. This strategy simplifies complex trigonometric equations and allows us to find the angle solutions efficiently.
In our textbook problem, \( \sin^2 x = 3 \cos x + 1 \), we leverage the relationship between the sine and cosine functions to find a solution. By applying the fundamental trigonometric identity \( \sin^2 x + \cos^2 x = 1 \), we can manipulate the equation and solve it in terms of a single trigonometric function—either sine or cosine. This strategy simplifies complex trigonometric equations and allows us to find the angle solutions efficiently.
Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the included variables. They are essential tools in solving trigonometric equations because they allow us to rewrite trigonometric expressions in different forms.
For example, the identity \( \sin^2 x + \cos^2 x = 1 \) is fundamental in transforming equations and making them solvable, as it binds the sine and cosine functions into a single, manageable expression. Other identities, such as double angle formulas or sum and difference formulas, extend our ability to work with and solve an even broader range of trigonometric problems.
Understanding and applying these identities can be the difference between recognizing a solution and being stumped by a complex equation. They provide the scaffolding necessary for manipulating and solving equations that, at first glance, may seem unsolvable.
For example, the identity \( \sin^2 x + \cos^2 x = 1 \) is fundamental in transforming equations and making them solvable, as it binds the sine and cosine functions into a single, manageable expression. Other identities, such as double angle formulas or sum and difference formulas, extend our ability to work with and solve an even broader range of trigonometric problems.
Understanding and applying these identities can be the difference between recognizing a solution and being stumped by a complex equation. They provide the scaffolding necessary for manipulating and solving equations that, at first glance, may seem unsolvable.
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