Problem 41 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 39-62: Find the solutions of the equation that are in the interval \([0,2 \pi\) ). $$ 2-8 \cos ^{2} t=0 $$

Step-by-Step Solution

Answer

The solutions are \(t = \frac{\pi}{3}, \frac{5\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}\) within \([0, 2\pi)\).

Step 1: Simplify the Equation

The given equation is \(2 - 8 \cos^2 t = 0\). First, isolate the \(\cos^2 t\) term by moving 2 to the other side: \[-8 \cos^2 t = -2\].Now, divide both sides by -8 to solve for \(\cos^2 t\): \[\cos^2 t = \frac{2}{8} = \frac{1}{4}\].

Step 2: Solve for \(\cos t\)

Take the square root of both sides of the equation to find \(\cos t\). Be sure to consider both the positive and negative roots:\[\cos t = \pm \frac{1}{2}\].

Step 3: Find the Values of \(t\)

Using the unit circle, find the values of \(t\) where \(\cos t = \frac{1}{2}\) and \(\cos t = -\frac{1}{2}\).For \(\cos t = \frac{1}{2}\):- \(t = \frac{\pi}{3}\)- \(t = \frac{5\pi}{3}\)For \(\cos t = -\frac{1}{2}\):- \(t = \frac{2\pi}{3}\)- \(t = \frac{4\pi}{3}\)

Step 4: Verify the Interval

Check each solution to ensure it fits within the interval \([0, 2\pi)\):- \(\frac{\pi}{3}\), \(\frac{5\pi}{3}\), \(\frac{2\pi}{3}\), and \(\frac{4\pi}{3}\) all fall within this interval.

Key Concepts

Unit Circle

The unit circle is a circle with radius 1, centered at the origin of an x-y coordinate plane. It plays a crucial role in trigonometry because it provides a simple way to visualize and define the trigonometric functions. The circle's circumference represents angles measured in radians, where one full rotation is equal to \(2\pi\) radians.
Every point on the unit circle corresponds to a cosine and sine value. These are the x- and y-coordinates of that point, respectively. So, if you know the angle, you can find the cosine by looking at the x-coordinate of the corresponding point on the circle.
This tool is especially handy for finding angles that satisfy equations like \(\cos t = \pm \frac{1}{2}\). Using symmetry and known angle values on the unit circle, you can pinpoint where the cosine equals specific values across different quadrants.

Cosine Function

The cosine function is one of the primary trigonometric functions, relating the angle of a right triangle to the ratio of the adjacent side to the hypotenuse. It is periodic and oscillates between -1 and 1.
This function is even, meaning that \(\cos(-t) = \cos(t)\), which reflects its symmetry about the y-axis. In unit circle terms, the cosine function corresponds to the x-coordinate of a point on the unit circle.
In solving trigonometric equations, identifying values of \(t\) where \(\cos t\) is either \( \frac{1}{2} \) or \(-\frac{1}{2}\) is essential. This can be achieved by considering well-known unit circle angles or using the cosine function's properties, like its range \([-1, 1]\). This allows for efficient solutions when combined with the unit circle.

Solution Verification

After solving a trigonometric equation, it's paramount to verify the solutions to ensure they are correct and fall within the designated interval, in this case, \([0, 2\pi)\).
Verification involves:

• Re-computing each solution to see if substituting back into the original equation equals zero.
• Checking if all solutions indeed lie within the specified interval by comparing the calculated angles (such as \(\frac{\pi}{3}\), \(\frac{5\pi}{3}\), etc.) against the interval boundaries.

Proper verification not only confirms the correctness but also that no extraneous solutions (solutions not applicable within the given constraints) are included. It ensures a reliable and complete solution set.

Interval Notation

Interval notation is a concise way of expressing a range of values. It is especially useful in mathematics for indicating solutions within specific bounds.
This notation denotes the interval using brackets:

• \([a, b]\) includes all values between \(a\) and \(b\), including \(a\) and \(b\) themselves.
• \((a, b)\) includes all values between \(a\) and \(b\), excluding \(a\) and \(b\).
• \([a, b)\) or \((a, b]\) indicate half-open intervals where only one boundary is included.

In the given exercise, the interval \([0, 2\pi)\) is a half-open interval, starting at 0 (inclusive) and ending before \(2\pi\) (not inclusive). Understanding this helps in confirming which solutions fall within a questioned range, ensuring all solutions respect the specified bounds.