Problem 37 - Contemporary Precalculus

Question

Use the graphs of the trigonometric functions to determine the number of solutions of the equation between 0 and \(2 \pi\) \(\cos t=k,\) where \(k\) is a constant such that \(-1

Step-by-Step Solution

Answer

Answer: 2 solutions

Step 1: Graph of the cosine function

The cosine function, \(\cos t\), has a period of \(2\pi\), with a maximum value of 1 and a minimum value of -1. Within one period, the graph is symmetric about the vertical line \(t=\pi\). The graph starts at a maximum, decreases to its minimum and then back to its maximum within one period.

Step 2: Determine the possible solutions

Given that \(-1

Step 3: Conclusion

In conclusion, for any value of \(k\) such that \(-1

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos t \), is one of the fundamental trigonometric functions. It represents the X-coordinate of a point on the unit circle as it travels around the circle. This particular function begins with a value of 1 when \( t = 0 \). As \( t \) increases from 0 to \( \pi \), the cosine function decreases to -1, reaching its minimum.

• Maximum Value: 1
• Minimum Value: -1

The behavior of the cosine function is crucial for understanding phenomena based on wave patterns, such as sound and light waves. Its smooth, continuous nature makes it a powerful tool in both mathematics and engineering.The cosine function is intimately linked with other trigonometric functions such as sine and tangent, and plays a vital role when solving equations like \( \cos t = k \).
The nature of this function's graph helps us visually see how many solutions exist for given values.

Graph of a Function

The graph of the cosine function is a wave-like pattern that repeats itself at regular intervals. This pattern is what defines its graph as a wave or sinusoidal. It provides a visual cue for how the function behaves over a range of values.

• Appearance: Starts at a peak (1), dips to a trough (-1), and then back to a peak.
• Symmetry: The graph is symmetric about vertical line \( t = \pi \).

Understanding the graph involves knowing its key characteristics:- **Amplitude:** The height from the middle of the wave to its peak is constant.- **Axis:** While it oscillates between -1 and 1, the midpoint or axis is at 0 (the horizontal line).Each repeat of the wave pattern corresponds to one full period of the function.

Periodicity

Periodicity is a critical concept in trigonometry that describes how often a function repeats its values. For the cosine function, periodicity illustrates its cyclical nature over a specific interval.

• Period: The cosine function has a period of \( 2\pi \). This means the entire cycle completes every \( 2\pi \) units.
• Regular Behavior: Within this period, it starts at a maximum, reaches a minimum, and returns to its maximum, maintaining a consistent shape.

When solving equations like \( \cos t = k \), periodicity helps determine how many solutions exist within a given interval. For each cycle from 0 to \( 2\pi \), if the solution is within the range of the function's values (-1 to 1 as in this case with \( -1 < k < 1 \)), there are two intersection points, indicating two solutions per cycle. Recognizing the repeating pattern simplifies finding these solutions.

Previous

Problem 37

Next

Problem 37

Other exercises in this chapter

Problem 36

In Exercises \(31-36\), write the expression as a single real number. Do not use decimal approximations. IHint: Exercises \(15-21 \text { may be helpful. }]\) $

View solution

Problem 37

Graph \(f(t)\) in a viewing window with \(-2 \pi \leq t \leq 2 \pi .\) Use a maximum finder and a root finder to determine constants \(A, b, c\) such that the g

View solution

Problem 37

Assume that $$\cos t=-2 / 5 \quad \text { and } \quad \pi

View solution

Problem 37

In Exercises \(37-42\), factor and simplify the given expression. $$\sec t \csc t-\csc ^{2} t$$

View solution

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos t \), is one of the fundamental trigonometric functions. It represents the X-coordinate of a point on the unit circle as it travels around the circle. This particular function begins with a value of 1 when \( t = 0 \). As \( t \) increases from 0 to \( \pi \), the cosine function decreases to -1, reaching its minimum.

• Maximum Value: 1
• Minimum Value: -1

The behavior of the cosine function is crucial for understanding phenomena based on wave patterns, such as sound and light waves. Its smooth, continuous nature makes it a powerful tool in both mathematics and engineering.The cosine function is intimately linked with other trigonometric functions such as sine and tangent, and plays a vital role when solving equations like \( \cos t = k \).
The nature of this function's graph helps us visually see how many solutions exist for given values.

Graph of a Function

The graph of the cosine function is a wave-like pattern that repeats itself at regular intervals. This pattern is what defines its graph as a wave or sinusoidal. It provides a visual cue for how the function behaves over a range of values.

• Appearance: Starts at a peak (1), dips to a trough (-1), and then back to a peak.
• Symmetry: The graph is symmetric about vertical line \( t = \pi \).

Understanding the graph involves knowing its key characteristics:- **Amplitude:** The height from the middle of the wave to its peak is constant.- **Axis:** While it oscillates between -1 and 1, the midpoint or axis is at 0 (the horizontal line).Each repeat of the wave pattern corresponds to one full period of the function.

Periodicity

Periodicity is a critical concept in trigonometry that describes how often a function repeats its values. For the cosine function, periodicity illustrates its cyclical nature over a specific interval.

• Period: The cosine function has a period of \( 2\pi \). This means the entire cycle completes every \( 2\pi \) units.
• Regular Behavior: Within this period, it starts at a maximum, reaches a minimum, and returns to its maximum, maintaining a consistent shape.

When solving equations like \( \cos t = k \), periodicity helps determine how many solutions exist within a given interval. For each cycle from 0 to \( 2\pi \), if the solution is within the range of the function's values (-1 to 1 as in this case with \( -1 < k < 1 \)), there are two intersection points, indicating two solutions per cycle. Recognizing the repeating pattern simplifies finding these solutions.