Question

Solve the equation for \(0 \leq x<2 \pi\). \(\tan \left(x-\frac{\pi}{4}\right)=0\)

Short answer

The solutions to the equation \(\tan \left(x-\frac{\pi}{4}\right)=0\) in the range \(0 \leq x < 2\pi\) are \(x = \frac{\pi}{4}, \frac{5\pi}{4}\).

Step-by-step solution

Identify the Form of the Equation

We are trying to solve \(\tan \left(x-\frac{\pi}{4}\right) = 0\). Remembering that \(\tan(\theta) = 0\) for \(\theta = n\pi\), we therefore have that \(x-\frac{\pi}{4} = n\pi\)

Solve for x

To find \(x\), add \(\frac{\pi}{4}\) to both sides of the equation, which gives \(x= n\pi + \frac{\pi}{4}\). As \(x\) is in the range \(0 \leq x < 2\pi\), we only have to test values of \(n\) which land \(x\) into that range.

Substitute Possible n-values

Substitute \(n = 0\) into \(x=n\pi + \frac{\pi}{4}\), get \(x_1 = \frac{\pi}{4}\). Now substitute \(n = 1\) and get \(x_2 = \pi + \frac{\pi}{4} = \frac{5\pi}{4}\). When \(n = 2\), \(x = 2\pi + \frac{\pi}{4} = \frac{9\pi}{4}\), which is not in the desired range. So the solutions in the range \(0 \leq x < 2\pi\) are \(\frac{\pi}{4}\) and \(\frac{5\pi}{4}\).

Key concepts

Tangent Function

The tangent function, often represented as \( \tan(\theta) \), is one of the primary trigonometric functions. It is defined as the ratio of the sine and cosine functions, that is, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). This means the value of \( \tan(\theta) \) is undefined wherever \( \cos(\theta) = 0 \), which occurs at odd multiples of \( \frac{\pi}{2} \).

The tangent function has some interesting properties:

• It has a period of \( \pi \), meaning \( \tan(\theta + \pi) = \tan(\theta) \).
• Tangent function intercepts the x-axis (has a value of zero) at integer multiples of \( \pi \), i.e., where \( \theta = n\pi \) for any integer \( n \).
• \( \tan(\theta) \) is symmetric about the origin, showing odd symmetry on its graph.

Being able to identify when \( \tan(\theta) = 0 \) is crucial for solving equations involving the tangent function. As such, understanding these properties can help simplify solving trigonometric equations.

Angle Rotation

Angle rotation is an essential concept when solving trigonometric equations because the angles are often given in terms of radians or degrees. Here, we're dealing with radians.

When thinking about angle rotation:

• The full circle in radians is \( 2\pi \), similar to how a full circle in degrees is \( 360^\circ \).
• A half rotation, or a semicircle, corresponds to \( \pi \) radians, equivalent to \( 180^\circ \).
• Quarter and three-quarter rotations are represented by \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \) respectively.

In solving our equation \( \tan \left(x-\frac{\pi}{4}\right) = 0 \), the idea of angle rotation helps in understanding how to modify the angle "\(x\)" based on the equation we derived: \( x = n\pi + \frac{\pi}{4} \).

Each solution is a rotation from the base angle, talking in terms of \( \pi \) lets us "jump" in specific amounts (multiples of \( \pi \)) to find valid rotations that lie within the specified interval \( 0 \leq x < 2\pi \). Hence, carefully considering how to rotate around the circle ensures the solutions remain in the correct specified range.

Principal Value of Trigonometric Functions

The principal value of a trigonometric function helps in finding the most intuitive solution to trigonometric equations by identifying a unique value for the angle that satisfies the equation. For the tangent function, the principal value is typically chosen within the interval \( -\frac{\pi}{2} < \theta \leq \frac{\pi}{2} \). This is because within this interval, the tangent function covers all possible values from negative infinity to positive infinity.

When dealing with the given exercise, our derived equation \( x = n\pi + \frac{\pi}{4} \) provides solutions within the interval \( 0 \leq x < 2\pi \). This larger interval is manageable given tangent's periodicity and helps ensure we find all possible solutions within the specified range.

For equations involving trigonometric functions:

• Identifying principal values helps achieve unique and specific answers.
• The principal value serves as a starting point for detecting other equivalent angles by leveraging trigonometric periodic properties.

By considering the principal value for tangent specifically, we can confidently obtain all necessary solutions without redundancy, streamlining the problem-solving process.