Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-50: Refer to the graph of \(y=\tan x\) to find the exact values of \(x\) in the interval \((-\pi / 2,3 \pi / 2)\) that satisfy the equation. $$ \tan x=\sqrt{3} $$

Step-by-Step Solution

Answer

The values of \(x\) are \(\frac{\pi}{3}\) and \(\frac{4\pi}{3}\).

Step 1: Recall the Tangent Function Properties

The tangent function, \(y = \tan x\), has a period of \(\pi\). This means the function repeats every \(\pi\) units. It is also undefined at certain intervals (specifically at \(x = \frac{\pi}{2} + k\pi\) for any integer \(k\)).

Step 2: Identify Known Tangent Angles

The value \(\tan x = \sqrt{3}\) is a known tangent value. It corresponds to angles where \(x = \frac{\pi}{3}\) and any angle coterminal with it, namely, \(x = \frac{\pi}{3} + k\pi\) for any integer \(k\).

Step 3: Determine the Interval

The task is to find \(x\) within the interval \((-\frac{\pi}{2}, \frac{3\pi}{2})\). This interval represents one and a half periods of the tangent function's cycle.

Step 4: Solve for \(x\) in the Given Interval

Since \(x = \frac{\pi}{3}\) is within the interval, it satisfies the equation. We need to consider \(\frac{\pi}{3} + k\pi\) for values of \(k\) such that \(x\) remains within the interval. Hence, \(k = 0\) gives \(x = \frac{\pi}{3}\), and \(k = 1\) gives \(x = \frac{\pi}{3} + \pi = \frac{4\pi}{3}\). Both \(\frac{\pi}{3}\) and \(\frac{4\pi}{3}\) are in the interval \((-\frac{\pi}{2}, \frac{3\pi}{2})\).

Step 5: List the Solutions

The exact values of \(x\) that satisfy \(\tan x = \sqrt{3}\) in the interval \((-\frac{\pi}{2}, \frac{3\pi}{2})\) are \(x = \frac{\pi}{3}\) and \(x = \frac{4\pi}{3}\).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental mathematical functions that are widely used in geometry, physics, engineering, and other fields. They describe relationships between angles and sides in right triangles. The basic trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each function is defined based on the properties of a right triangle:

• Sine: \(\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}\)
• Cosine: \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\)
• Tangent: \(\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}\)

The tangent function specifically is very useful for calculating the slope of a line and various engineering calculations. It's important to understand that as \(\theta\) approaches certain values, \(\tan(\theta)\) becomes undefined. This occurs because the tangent function involves division by the cosine function, which can be zero at certain angles. Knowing the basic properties of these functions is crucial when solving trigonometric equations.

Periodicity of Functions

The concept of periodicity in functions refers to the regular and repetitive nature with which some functions occur. In the context of the tangent function, \(y = \tan(x)\), this periodicity is particularly crucial. The tangent function has a period of \(\pi\). This means that the function repeats its values every \(\pi\) units along the x-axis.

Periodic functions like \(\tan(x)\) can be visualized in a graph as repeating waves or cycles. In the case of \(\tan(x)\), we see the pattern recurring without end, indicating the positions where the function becomes undefined. These points occur at \(x = \frac{\pi}{2} + k\pi\) for any integer \(k\).

Understanding periodicity helps when solving inequalities or equations involving tangent because it allows us to predict and map the recurring solutions. Knowing the period of a function allows us to find all the possible solutions over a given interval, as in the problem we are considering.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides of the equation are defined. These identities allow us to simplify complex trigonometric expressions and solve trigonometric equations more efficiently. For the tangent function, an important identity is its relationship with sine and cosine:\[\tan(x) = \frac{\sin(x)}{\cos(x)}\]This identity indicates that if \(\cos(x)\) equals zero, \(\tan(x)\) is undefined, which aligns with the periodicity and the function's graph.

Other helpful identities include the Pythagorean identity, which is primarily for sine and cosine but helps explain the properties of all trigonometric functions:\[\sin^2(x) + \cos^2(x) = 1\]Such identities are essential tools in both simplifying expressions and solving equations related to trigonometric functions. They form the backbone of understanding how these functions behave and how they interrelate. By mastering these identities, students can find solutions to otherwise complex trigonometric equations, like finding when \(\tan(x) = \sqrt{3}\) in specific intervals.