Problem 28 - Calculus

Question

Find every \(\theta\) that satisfies the equation. $$ \tan \theta=0 $$

Step-by-Step Solution

Answer

\( \theta = n\pi \), where \( n \) is an integer.

Step 1: Understand the Tangent Function

The tangent function, represented as \( \tan \theta \), is the ratio of the sine and cosine of an angle. It is periodic, repeating every \( \pi \) radians.

Step 2: Set Up the Equation

We are given \( \tan \theta = 0 \). This means we're looking for angles where the tangent, which is \( \frac{\sin \theta}{\cos \theta} \), is zero.

Step 3: Determining When Tangent is Zero

The tangent function is zero when \( \sin \theta = 0 \). The sine of an angle is zero at integer multiples of \( \pi \), i.e., \( \theta = n\pi \) where \( n \) is any integer.

Step 4: General Solution

Since the tangent function is periodic with period \( \pi \), all solutions to the equation \( \tan \theta = 0 \) are of the form \( \theta = n\pi \).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics, particularly in understanding angles and their relationships in triangles. The primary trigonometric functions are sine (\sin), cosine (\cos), and tangent (\tan). Each function relates a particular angle to specific ratios of a right triangle's sides.
The tangent function, \( \tan \theta \), is a unique one because it can be represented as the ratio of sine to cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
This means that whenever the cosine of an angle is zero, the tangent function becomes undefined, since you cannot divide by zero. However, the tangent function is zero when the sine of an angle is zero.

• The angles where the sine function equals zero are critical in determining when the tangent function equals zero.
• Learning these foundational functions helps solve a wide range of mathematical problems and understand how angles behave.

Periodicity

The periodicity of a function refers to how often the function repeats its values over intervals. For the tangent function, periodicity is a key characteristic. Tangent is periodic with a period of \( \pi \) radians.
This means that every \( \pi \) radians, the tangent function's values start repeating. For example, if \( \tan \theta = 0 \) at \( \theta = 0 \), it will also be zero at \( \theta = \pi \), \( 2\pi \), \( 3\pi \), and so on.
The concept of periodicity impacts how we find solutions to equations involving trigonometric functions:

• The knowledge that the function repeats at regular intervals allows us to foresee an infinite set of solutions for the equation \( \tan \theta = 0 \).
• This repeated pattern makes it easier to predict and understand angles that satisfy given trigonometric equations.

Solution to Equations

Solving equations involving trigonometric functions often involves finding all possible angles or values that satisfy a given condition. In our problem, we aim to solve \( \tan \theta = 0 \).
Since the tangent function \( \tan \theta \) equals zero when the sine of \( \theta \) is zero, we're essentially looking for the zeros of the sine function, which occur at multiples of \( \pi \).

• The general solution for \( \tan \theta = 0 \) is \( \theta = n\pi \), where \( n \) is any integer.
• This comprehensive approach takes advantage of the function's periodicity to identify all solutions.

Understanding these ideas is not only crucial in solving equations like the one given but also builds a foundation for more complex trigonometric and analytical problems.