Problem 14 - Algebra and Trigonometry

Question

Find all solutions of each equation. $$ \tan x=\sqrt{3} $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \pi / 3 + n\pi\), where \(n\) is any integer.

Step 1: Identifying Value of Tangent

From trigonometry, we know that the value of \(\tan x\) is \(\sqrt{3}\) for \(x=\pi / 3\) in the domain \([0, 2\pi]\). This is obtained from the unit circle or the 30-60-90 triangle.

Step 2: Applying Periodicity of Tangent

The function \(\tan x\) is periodic with period \(\pi\). Therefore, for any integer \(n\), the function will take the same value for \(x = \pi / 3 + n\pi\). This gives us the infinite set of solutions for the equation.

Key Concepts

Tangent Function

The tangent function is a fundamental concept in trigonometry, related to the ratio of the sides of a right-angled triangle. Specifically, for an angle \( x \), the tangent of \( x \) is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle.

When it comes to the unit circle representation, the tangent function can be thought of as the length of the line segment that is tangent to the unit circle at the point \( (1,0) \) and intersects the line that makes an angle \( x \) with the positive x-axis. This geometric interpretation helps explain why the tangent function can have any real number as its value, unlike the sine and cosine functions that are limited between \(-1\) and \(1\).

Understanding how the tangent function behaves for different angles is crucial. For example, when we say that \( \tan x = \sqrt{3} \), it refers to angles where the opposite side is \( \sqrt{3} \) times longer than the adjacent side. These angles are typically found at 60 degrees (or \( \pi / 3 \) radians) and their coterminal angles. This is important to grasp because solving trigonometric equations often involves finding all the angles that satisfy a particular tangent value.

Unit Circle

The unit circle is an essential tool in trigonometry, representing all the possible angles and the values of trigonometric functions. The circle has a radius of 1 unit, and is centered at the origin (0,0) of the coordinate plane.

The beauty of the unit circle is that any point \( (x,y) \) on the circumference corresponds to an angle with a cosine value of \( x \) and a sine value of \( y \). The tangent value can be understood as the ratio \( y/x \) for any point on the circle.

Importance of the Unit Circle in Solving Trigonometric Equations

When solving equations like \( \tan x = \sqrt{3} \), we use the unit circle to identify the base angle - in this case, \( x = \pi / 3 \). However, because tangent has symmetry, for each positive angle with a certain tangent value, there is a negative angle with the same tangent value (but opposite in sign). The unit circle helps visualize this symmetry and guides us to all possible solutions of a tangent equation.

Periodicity of Trigonometric Functions

Periodicity is a property of trigonometric functions that indicates the functions repeat their values in regular intervals. In essence, if you measure a trigonometric function at an interval equal to its period, the value will be the same as it was at the start of that interval.

For the tangent function specifically, the period is \( \pi \) radians. This means that \( \tan(x + n\pi) = \tan(x) \), for any integer \( n \).

Using Periodicity to Solve Equations

If you determine that \( \tan(x) = \sqrt{3} \) for some value of \( x \), due to periodicity, you can find infinitely many solutions by adding multiples of \( \pi \) to the base solution. In our exercise, the equation \( \tan x = \sqrt{3} \) has a base solution of \( x = \pi / 3 \), but thanks to periodicity, additional solutions are found by adding integer multiples of the period \( \pi \), hence the general solution is \( x = \pi / 3 + n\pi \).

Appreciating the concept of periodicity is crucial for students as it ensures a comprehensive understanding of trigonometric equations and aids in identifying all possible solutions.