Problem 5 - Contemporary Precalculus

Question

Use the graphs of the sine and cosine functions to find all the solutions of the equation. $$\cos t=-1$$

Step-by-Step Solution

Answer

Question: Find all the solutions of the equation \(\cos t = -1\) using the graphs of the sine and cosine functions. Answer: The solutions for the equation \(\cos t = -1\) can be expressed in the following general form: \(t = \pi + 2n\pi, \quad n \in \mathbb{Z}\), where \(n\) is any integer value.

Step 1: Identify the period of the cosine function

The cosine function has a period of \(2\pi\), which means that its graph repeats every \(2\pi\) units. This will help us find the pattern for all the solutions of the given equation.

Step 2: Locate the first point where \(\cos t = -1\)

Recall that the cosine function has a range of \([-1,1]\). The cosine function takes a value of \(-1\) when \(t= \pi\). Therefore, the first solution is \(t = \pi\).

Step 3: Find the pattern for the other solutions

Since cosine has a period of \(2\pi\), we can find the rest of the solutions by adding integer multiples of \(2\pi\) to the first solution we found. Mathematically, we can express this as follows: $$t = \pi + 2n\pi, \quad n \in \mathbb{Z}$$ Where \(n\) is any integer value.

Step 4: Provide the general form for all solutions

Based on the pattern we found, we can express all the solutions of the equation \(\cos t = -1\) in the following general form: $$t = \pi + 2n\pi, \quad n \in \mathbb{Z}$$

Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions, often denoted as \( \cos \). It relates an angle with the ratio of the adjacent side to the hypotenuse in a right triangle.
The function is periodic, meaning it repeats its values in regular intervals. For cosine, these intervals are every \( 2\pi \) radians.

• The range of the cosine function is \([-1, 1]\). This means that for any angle \(t\), the value of \(\cos t\) will never exceed 1 or drop below -1.
• Graphically, the cosine function forms a wave-like pattern, starting at the maximum value of 1 at \(t = 0\), decreasing to -1, before rising back to 1.

To understand where \(\cos t = -1\), consider this wave pattern. It hits -1 at specific points, such as when \(t = \pi\), and because it is periodic, it keeps hitting -1 after each \(2\pi\) rotation.

Periodicity

Periodicity in trigonometric functions is the concept that these functions repeat their values in a predictable, regular interval. For the cosine function, this interval is precisely \(2\pi\) radians.
This means that starting from any angle \(t\), adding \(2\pi\) will bring you back to the same point on the cosine wave. The function looks the same at \(t\), \(t + 2\pi\), \(t + 4\pi\), and so on.

• This periodic nature is crucial in solving trigonometric equations as it allows us to express infinite solutions compactly.
• In our case, finding the solutions where \(\cos t = -1\) gives the base solution \(t = \pi\). Adding \(2n\pi\) (where \(n\) is an integer) to this solution captures all points where \(\cos t = -1\) again.

Overall, the topic of periodicity helps in understanding the repeating behavior of trigonometric functions and plays a vital role in solving equations involving them.

Trigonometric Equations

Trigonometric equations involve finding angles that satisfy a particular trigonometric condition. In our exercise, the equation \(\cos t = -1\) means you need to find all angles \(t\) where the cosine function equals -1.
Solving such equations requires understanding both the behavior and patterns of the trigonometric function involved.

• The basic solutions occur at specific known points (like \(t = \pi\) for cosine). These are derived from the graph and properties of the function.
• To get a general form of the solution, utilize the periodicity of the function, allowing you to express repeated solutions succinctly.

For instance, the general solution to \(\cos t = -1\) is expressed as \(t = \pi + 2n\pi, \quad n \in \mathbb{Z}\), which captures all possible angles \(t\) that satisfy the equation. This approach of working through the problem ensures you find every possible solution.