Problem 157 - Calculus Volume 1

Question

Solve the trigonometric equations on the interval \(0 \leq \theta<2 \pi\). \(2 \tan ^{2} \theta=2\)

Step-by-Step Solution

Answer

Solutions are \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\).

Step 1: Simplify the Equation

Start by simplifying the given equation. The original equation is \(2 \tan^2 \theta = 2\). Divide both sides by 2 to get \(\tan^2 \theta = 1\).

Step 2: Solve for Basic Solutions

Take the square root of both sides to find solutions for \(\theta\). As \(\tan^2 \theta = 1\), \(\tan \theta\) can be \(1\) or \(-1\). So, \(\tan \theta = 1\) or \(\tan \theta = -1\).

Step 3: Determine Angles for \(\tan \theta = 1\)

For \(\tan \theta = 1\), look for angles in the unit circle where this is true. This occurs at \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{5\pi}{4}\).

Step 4: Determine Angles for \(\tan \theta = -1\)

For \(\tan \theta = -1\), this occurs at angles \(\theta = \frac{3\pi}{4}\) and \(\theta = \frac{7\pi}{4}\).

Step 5: Verify the Solutions

Verify all solutions by plugging the angle back into the original equation. All identified \(\theta\) satisfy \(2 \tan^2 \theta = 2\) as each gives \(\tan^2 \theta = 1\).

Step 6: List All Solutions

Hence, the solutions within the interval \(0 \leq \theta < 2\pi\) are \(\theta = \frac{\pi}{4}\), \(\theta = \frac{3\pi}{4}\), \(\theta = \frac{5\pi}{4}\), and \(\theta = \frac{7\pi}{4}\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It's a circle with a radius of 1 centered at the origin of a coordinate plane. This circle is a powerful tool for understanding trigonometric functions, especially sine, cosine, and tangent.

• The unit circle provides a visual way to see the values of trigonometric functions at various angles, usually measured in radians.
• For angles, it's common to see measurements like \(\pi/4\), \(\pi/2\), \(\pi\), and so forth. These are specific points on the unit circle where the trigonometric functions have well-known values.
• In the context of the problem, solving \(\tan \theta = 1\) or \(\tan \theta = -1\) involves finding the points on the unit circle where the tangent of the angle returns these values.

Understanding where these values occur on the unit circle helps in quickly solving trigonometric equations. In this case, it helps determine exact angles \(\frac{\pi}{4}\), \(\frac{5\pi}{4}\), \(\frac{3\pi}{4}\), and \(\frac{7\pi}{4}\) for which the tangent function meets the required conditions.

Tangent Function

The tangent function is one of the primary trigonometric functions. It can be defined as the ratio of the sine and cosine functions:\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]This definition makes the tangent function unique in its behavior and properties.

• The tangent function is periodic with a period of \(\pi\). This means it repeats its values every \(\pi\) radians.
• When the tangent is positive, both sine and cosine share the same signs. When negative, they differ in signs.
• In standard position, tangent is positive in the first and third quadrants and negative in the second and fourth quadrants of the Cartesian plane.

In solving the equation \(\tan^2 \theta = 1\), we determine that \(\tan \theta = 1\) or \(\tan \theta = -1\). This simplifies identifying specific angles in certain quadrants where the tangent function achieves these values.

Solutions Verification

Verification of solutions is a critical step in solving equations to ensure the identified values are indeed correct. It involves checking the solutions by substituting them back into the original equation to see if it holds true.

• For this problem, once we have calculated the angles \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\), we substitute each back into the equation \(2 \tan^2 \theta = 2\).
• After substitution, we simplify to test \(\tan^2 \theta\). Each angle should result in \(\tan^2 \theta = 1\), confirming consistency with the original equation.

This step ensures that no arithmetic mistakes were made and that these are indeed the actual solutions for the interval \(0 \leq \theta < 2\pi\). Confirming solutions solidifies understanding and correctness of problem-solving techniques.