Problem 58 - Precalculus: Functions and Graphs

Question

Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. $$3 \tan ^{4} \theta-19 \tan ^{2} \theta+2=0 ; \quad\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The solutions are approximately \( \theta = 1.1550, -1.1550, 0.3446, -0.3446 \).

Step 1: Identify Substitution

Notice that the equation is quadratic in form for \( an^2 \theta \). Let \( x = an^2 \theta \), then rewrite the original equation as \( 3x^2 - 19x + 2 = 0 \).

Step 2: Solve Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 3 \), \( b = -19 \), and \( c = 2 \). Calculating the discriminant gives \( b^2 - 4ac = 361 - 24 = 337 \). Hence, \( x = \frac{19 \pm \sqrt{337}}{6} \).

Step 3: Calculate Roots

Calculate the roots using the quadratic formula: \( x_1 = \frac{19 + \sqrt{337}}{6} \approx 5.1385 \) and \( x_2 = \frac{19 - \sqrt{337}}{6} \approx 0.1298 \).

Step 4: Find \( \tan \theta \)

Since \( x = \tan^2 \theta \), we need to find \( \tan \theta \). For \( x_1 = 5.1385 \), \( \tan \theta = \pm \sqrt{5.1385} \approx \pm 2.2668 \); for \( x_2 = 0.1298 \), \( \tan \theta = \pm \sqrt{0.1298} \approx \pm 0.3603 \).

Step 5: Determine \( \theta \) in Interval

Since we are given \( \theta \) in \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), we determine \( \theta = \tan^{-1}(2.2668) \approx 1.1550 \) and \( \theta = -\tan^{-1}(2.2668) \approx -1.1550 \), \( \theta = \tan^{-1}(0.3603) \approx 0.3446 \) and \( \theta = -\tan^{-1}(0.3603) \approx -0.3446 \).

Step 6: Verify Solutions within Interval

Verify each \( \theta \) to ensure it falls within \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \). All calculated angles are within this interval.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that is characterized by having the highest power of the unknown variable be two. This is typically expressed in the standard form as:\[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). The solutions to a quadratic equation can be found using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In this formula:

• \(b^2 - 4ac\) is called the discriminant.
• The discriminant helps determine the nature of the roots:
• If positive, the equation has two distinct real roots.
• If zero, it has exactly one real root (double root).
• If negative, the roots are complex or imaginary.

Understanding how to form and solve quadratic equations is crucial for solving problems involving quadratic expressions, such as those involving the tangent function squared.

Tangent Function

The tangent function, often abbreviated as \(\tan\), is a fundamental trigonometric function. It is defined as the ratio of the sine to the cosine of an angle in a right triangle:\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]The tangent function is periodic, with a period of \(\pi\), meaning it repeats its values every \(\pi\) units. It also has certain peculiar properties:

• The function is undefined wherever \(\cos(\theta) = 0\), which occurs at odd multiples of \(\frac{\pi}{2}\).
• The range of \(\tan(\theta)\) is all real numbers, as it can take any real value from \(-\infty\) to \(+\infty\).

When dealing with equations involving \(\tan^2(\theta)\), it is often useful to substitute \(x = \tan^2(\theta)\), as this can lead to a quadratic form that is easier to solve. This process helps in finding the angles corresponding to specific tangent values within a specified interval.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for every value of the involved variables. These identities are extremely useful in simplifying expressions and solving trigonometric equations. Some core trigonometric identities include:

• Pythagorean Identities:
  • \(\sin^2(\theta) + \cos^2(\theta) = 1\)
  • \(\tan^2(\theta) + 1 = \sec^2(\theta)\)
• Reciprocal Identities:: These define relations between trigonometric functions and their reciprocals.
  • \(\csc(\theta) = \frac{1}{\sin(\theta)}\)
  • \(\sec(\theta) = \frac{1}{\cos(\theta)}\)
  • \(\cot(\theta) = \frac{1}{\tan(\theta)}\)

Applying these identities can transform complex trigonometric equations into simpler forms, allowing easier computation of solutions, especially when involving inverse trigonometric functions.

Angle Interval

An angle interval specifies the range of angles for which we need to find solutions in a given trigonometric problem. When dealing with trigonometric equations, it is crucial to find solutions that lie within the desired angle interval to ensure they are viable in a given context.For instance, in this exercise, the interval is \((-\frac{\pi}{2}, \frac{\pi}{2})\). This interval is significant because:

• It defines the principal range of the inverse tangent function, \(\tan^{-1}(x)\) or \(\arctan(x)\).
• Within this range, the function \(\tan^{-1}(x)\) gives values for \(\theta\) such that the tangent function is bijective, meaning it has one distinct output for each input.

To verify solutions, ensure that all calculated \(\theta\) values fall within this interval, providing valid solutions to the trigonometric equation in question.