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Chapter 5: Problem 38

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\sin x-2=\cos x-2$$

Short Answer

Expert verified
The solutions to the equation \(\sin x-2= \cos x-2\) in the interval \([0,2\pi)\) are \(x=\frac{\pi}{4}\), \(x=\frac{5\pi}{4}\).

Step by step solution

01

Rewrite the equation

To simplify the equation, rewrite \(\sin x-2= \cos x-2\) as \(\sin x = \cos x\), because the \(-2\) cancels out from both sides of the equation.
02

Use the identity of sine and cosine

Since \(\sin x = \cos x\), these two functions are equal when this equation is satisfied: \(\frac{\pi}{4} + n\pi\), where \(n\) is any integer.
03

Determine x values within the interval [0, 2蟺)

The interval [0, 2蟺) indicates that the \(x\)-values should be equal to or more than 0 but less than 2蟺. Solving \(\frac{\pi}{4} + n\pi\) reveals that possible values within the demanded interval are \(x=\frac{\pi}{4}\), \(x=\frac{5\pi}{4}\).

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Key Concepts

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

Sine Function
The sine function is a trigonometric function that describes a smooth, periodic oscillation. It originates from the study of right-angled triangles and the circles. In a unit circle, the sine of an angle is the y-coordinate of the corresponding point:
  • For any angle \(x\), \( ext{sin}(x)\) ranges between -1 and 1.
  • The sine function is periodic with a period of \(2\pi\), meaning that it repeats every \(2\pi\) radians.
  • It is an odd function, characterized by the property: \(\sin(-x) = -\sin(x)\).
  • The graph of the sine function is a wave that starts at 0 when \(x = 0\), reaches 1 at \(x = \pi/2\), goes back to 0 at \(x = \pi\), -1 at \(x = 3\pi/2\), and completes a cycle at \(x = 2\pi\).
The equation \(\sin x = \cos x\) holds true where these values are equal, at points like \(x = \frac{\pi}{4} + n\pi\). Understanding the sine function's symmetry and periodicity helps in solving these equations efficiently.
Cosine Function
The cosine function complements the sine function and originates in similar geometric considerations on the unit circle. For any real number, the cosine function gives the x-coordinate of a corresponding point on the unit circle:
  • Just like the sine, \(\cos(x)\) ranges between -1 and 1.
  • The cosine function also has a periodic nature with a period of \(2\pi\).
  • Unlike sine, cosine is an even function, with the property: \(\cos(-x) = \cos(x)\).
  • The cosine function's graph begins at 1 when \(x = 0\), moves to 0 at \(x = \pi/2\), -1 at \(x = \pi\), 0 at \(x = 3\pi/2\), and completes a cycle as it returns to 1 at \(x = 2\pi\).
In the given trigonometric equation, the values where \(\sin x = \cos x\) hold important significance, highlighting the points where these periodic waves intersect, like \(x = \frac{\pi}{4} + n\pi\). Recognizing these positions requires understanding the cosine's peaks and symmetry.
Angle Interval
An angle interval in the context of trigonometric equations denotes the range in which we're searching for solutions. Here, the interval \[0, 2\pi)\] means that angles must satisfy:
  • Start at 0 radians, inclusive, which is equivalent to 0 degrees.
  • Go up to \(2\pi\) radians, exclusive, which corresponds to 360 degrees, completing a full circle without including the starting point again.
  • This kind of interval helps ensure that solutions are found without repeating previous cycle points.
  • The boundary conditions of this interval mean checking values exactly at the start and less than the given endpoint, e.g., values like \(x = \frac{\pi}{4}, \frac{5\pi}{4}\) ensure no overlap as above \(2\pi\) would start a new cycle.
Understanding this concept is crucial for correctly interpreting which solutions of a trigonometric equation lie within a desired range, aiding both the clarification and precision in solving equations like \(\sin x = \cos x\).

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Most popular questions from this chapter

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