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Chapter 6: Problem 11

Solve the given trigonometric equation exactly over the indicated interval. $$\tan (2 \theta)=\sqrt{3},-2 \pi \leq \theta<2 \pi$$

Short Answer

Expert verified
The solutions are \(-11\pi/6, -3\pi/2, -5\pi/6, -\pi/3, \pi/6, 2\pi/3, 7\pi/6, 11\pi/6\).

Step by step solution

01

Solve for the angle in radians

The equation is given by \( \tan(2\theta) = \sqrt{3} \). We know that \( \tan(\pi/3) = \sqrt{3} \). So, \( 2\theta = \pi/3 + k\pi \) for some integer \( k \). This is because the tangent function has a period of \( \pi \).
02

Solve for \( \theta \)

To find \( \theta \), divide both sides of the equation \( 2\theta = \pi/3 + k\pi \) by 2. This gives \( \theta = \pi/6 + k\pi/2 \), where \( k \) is an integer.
03

Find solutions within the interval

We need to find all \( \theta \) such that \(-2\pi \leq \theta < 2\pi \). To do this, evaluate \( \theta = \pi/6 + k\pi/2 \) for various integer values of \( k \). Temporarily calculating modest values of \( k \), we use \( k = -4, -3, -2, -1, 0, 1, 2, 3, 4 \). This generates potential solutions.
04

Evaluate integer values of \( k \)

For each \( k \), consider the following calculations:- \( k = -4 \), \( \theta = \pi/6 - 2\pi = -11\pi/6 \)- \( k = -3 \), \( \theta = \pi/6 - 3\pi/2 = -3\pi/2 + \pi/6 = -9\pi/6 = -3\pi/2 \)- \( k = -2 \), \( \theta = \pi/6 - \pi = -5\pi/6 \)- \( k = -1 \), \( \theta = \pi/6 - \pi/2 = -\pi/3 \)- \( k = 0 \), \( \theta = \pi/6 \)- \( k = 1 \), \( \theta = \pi/6 + \pi/2 = 2\pi/3 \)- \( k = 2 \), \( \theta = \pi/6 + \pi = 7\pi/6 \)- \( k = 3 \), \( \theta = \pi/6 + 3\pi/2 = 11\pi/6 \)
05

Ensure values are within the specified range

Check that each \( \theta \) value is within \(-2\pi \leq \theta < 2\pi \). The acceptable solutions are: \(-11\pi/6, -3\pi/2, -5\pi/6, -\pi/3, \pi/6, 2\pi/3, 7\pi/6, 11\pi/6 \). These are the final answers for \( \theta \) within the given range.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Function
The tangent function, denoted as \( \tan \), is a fundamental concept in trigonometry. It is one of the three primary trigonometric functions, the others being sine and cosine. The tangent of an angle \( \theta \) in a right triangle is the ratio of the length of the opposite side to the adjacent side. Moreover, in the unit circle, it is defined as the ratio of the y-coordinate to the x-coordinate for a point forming angle \( \theta \) with the positive x-axis.
  • The tangent function is periodic, with a period of \( \pi \) radians. This means \( \tan(\theta + \pi) = \tan(\theta) \).
  • It has an infinite range but is undefined whenever the cosine is zero because it involves division by zero.
  • The function produces every real number within its period, and it repeats itself.
In the context of solving equations like \( \tan(2\theta) = \sqrt{3} \), we leverage the periodicity to find multiple solutions over a given interval by adding \( k\pi \), where \( k \) is an integer.
Angle in Radians
Radians are a way of measuring angles, distinct from the more common degrees. In simple terms, an angle measured in radians creates an arc on a circle's circumference equal in length to the circle's radius. A complete circle has an angle of \( 2\pi \) radians.
  • 1 radian is approximately equal to 57.2958 degrees.
  • Commonly used exact values include \( \pi \) radians for a straight angle (180 degrees).
  • Radians provide a natural way to express periodic functions, like sine, cosine, and tangent.
For solving the equation \( \tan(2\theta) = \sqrt{3} \), we express solutions in radians, as angles of \( \pi/3 \) form the initial basis of the solutions, but it's important to recall radians provide a succinct way to handle circular functions.
Interval Notation
Interval notation is a mathematical notation system that describes a specific range of numbers between two endpoints. It is particularly helpful in describing domains for functions or formulating solution sets of inequalities and equations.
  • A closed interval \([a, b]\) includes both endpoints \(a\) and \(b\).
  • An open interval \((a, b)\) excludes the endpoints.
  • Mixed intervals \((a, b]\) or \([a, b)\) include one endpoint but not the other.
When solving trigonometric equations, such as \( \tan(2\theta) = \sqrt{3} \) over a stated interval, interval notation helps us clearly express the set of solutions. For this problem, \(-2\pi \leq \theta < 2\pi\) indicates we are looking for all possible angle values from \(-2\pi\) to just under \(2\pi\), ensuring we correctly identify all solutions available within this range.

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Most popular questions from this chapter

Write \(\sin A \sin B \sin C\) as a sum or difference of sines and cosines.

Assume that light is going from a diamond into air. Calculate the refractive angle \(\theta_{r}\) if the incidence angle is \(\theta_{i}=15^{\circ}\) and the index of refraction values for diamond and air are \(n_{i}=2.417\) and \(n_{r}=1.00,\) respectively. Round to the nearest degree. (See Example 12 for Snell's law.)

Graphing calculators can be used to find approximate solutions to trigonometric equations. For the equation \(f(x)=g(x),\) let \(Y_{1}=f(x)\) and \(Y_{2}=g(x) .\) The \(x\) -values that correspond to points of intersections represent solutions. With a graphing utility, find all of the solutions to the equation \(\cos \theta=e^{\theta}\) for \(\theta \geq 0\).

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