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Magazine Chapter 10: Problem 41
Solve each equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right) .\) Give solutions to the nearest tenth as appropriate. $$\tan ^{2} \theta+4 \tan \theta+2=0$$
Short Answer
Expert verified
The solutions are approximately \( \theta = 123.3^\circ \) and \( \theta = 303.3^\circ \).
Step by step solution
01
Identify the Equation Type
The given equation is quadratic in terms of \( \tan \theta \). It can be rewritten as \( x^2 + 4x + 2 = 0 \) where \( x = \tan \theta \).
02
Use the Quadratic Formula
To solve for \( x = \tan \theta \), we apply the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = 4 \), and \( c = 2 \).
03
Calculate the Discriminant
The discriminant \( b^2 - 4ac \) is \( 4^2 - 4 \times 1 \times 2 = 16 - 8 = 8 \).
04
Solve Using the Quadratic Formula
Substitute the discriminant into the quadratic formula: \[ x = \frac{-4 \pm \sqrt{8}}{2} = \frac{-4 \pm 2\sqrt{2}}{2} \] Simplifying gives: \[ x = -2 \pm \sqrt{2} \]
05
Find \( \theta \) using \( x = \tan \theta \)
Solving for \( \theta \) means finding \( \theta \) such that \( \tan \theta = -2 + \sqrt{2} \) or \( \tan \theta = -2 - \sqrt{2} \).
06
Calculate \( \theta \) for Each Solution
Convert each \( x \) value into an angle \( \theta \) using the inverse tangent function. - For \( \tan \theta = -2 + \sqrt{2} \), calculate \( \theta \) using \( \theta = \tan^{-1}(-2 + \sqrt{2}) \). - For \( \tan \theta = -2 - \sqrt{2} \), calculate \( \theta = \tan^{-1}(-2 - \sqrt{2}) \). Adjust the angles to be within \( [0^\circ, 360^\circ) \).
07
Adjust Solutions to Correct Interval
Evaluate the angles calculated from \( \tan^{-1} \) to ensure they lie between \( 0^\circ \) and \( 360^\circ \). If needed, use the periodicity of the tangent function (add 180掳 to each solution to find other possible solutions in the interval).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equations
Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). Solving a quadratic equation involves finding the values of \( x \) that satisfy the equation.
Common methods for solving quadratic equations include:
In trigonometric contexts, a quadratic equation often appears in terms of functions like \( \tan \theta \) or \( \sin \theta \), and solving them might involve additional steps like using inverse trigonometric functions.
Common methods for solving quadratic equations include:
- Factoring: Useful when the quadratic can be expressed as a product of two binomials.
- Quadratic Formula: Given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which provides the solution in terms of \( a \), \( b \), and \( c \).
- Completing the square: A technique that involves rewriting the equation so one side is a perfect square trinomial.
In trigonometric contexts, a quadratic equation often appears in terms of functions like \( \tan \theta \) or \( \sin \theta \), and solving them might involve additional steps like using inverse trigonometric functions.
Tangent Function
The tangent function, denoted as \( \tan \theta \), is one of the fundamental trigonometric functions, alongside sine and cosine. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side.
In terms of the unit circle, \( \tan \theta \) can be expressed as \( \frac{\sin \theta}{\cos \theta} \), and its value can vary from negative to positive infinity.
Understanding the properties of the tangent function helps when finding solutions to equations like \( \tan \theta = -2 + \sqrt{2} \) by predicting the behavior and possible angles in a given interval.
In terms of the unit circle, \( \tan \theta \) can be expressed as \( \frac{\sin \theta}{\cos \theta} \), and its value can vary from negative to positive infinity.
- Periodicity: The tangent function is periodic with a period of \( 180^{\circ} \) or \( \pi \, \text{radians} \).
- Undefined Points: The function is undefined when \( \cos \theta = 0 \), corresponding to angles like \( 90^{\circ} \) or \( 270^{\circ} \).
Understanding the properties of the tangent function helps when finding solutions to equations like \( \tan \theta = -2 + \sqrt{2} \) by predicting the behavior and possible angles in a given interval.
Inverse Trigonometric Functions
Inverse trigonometric functions allow the determination of angles from given trigonometric values. The inverse function for tangent is denoted \( \tan^{-1} \) or \( \arctan \), which returns the angle \( \theta \) whose tangent is a given number.
It is important to remember that since \( \tan \theta \) is periodic:
In practical applications, like in the given exercise, it is crucial to adjust the solutions obtained using \( \tan^{-1} \) to the desired interval, which is \( [0^{\circ}, 360^{\circ}) \). This often requires adding or subtracting \( 180^{\circ} \) to fit the angle within the specified range.
It is important to remember that since \( \tan \theta \) is periodic:
- \( \tan^{-1}(x) \) typically provides angles between \( -90^{\circ} \) and \( 90^{\circ} \).
- Multiple solutions may exist due to the function's periodic nature. These solutions differ by multiples of \( 180^{\circ} \).
In practical applications, like in the given exercise, it is crucial to adjust the solutions obtained using \( \tan^{-1} \) to the desired interval, which is \( [0^{\circ}, 360^{\circ}) \). This often requires adding or subtracting \( 180^{\circ} \) to fit the angle within the specified range.
