Problem 94 - Algebra and Trigonometry

Question

Use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$ \tan ^{2} x-3 \tan x+1=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \tan^{-1}(\frac{3 + \sqrt{5}}{2})\) and \(x = \tan^{-1}(\frac{3 - \sqrt{5}}{2})\), provided that these solutions fall within the interval [0, 2π). You will need to use a calculator to get the exact decimal values for the solutions.

Step 1: Solve the quadratic equation

The given equation \(\tan^2 x - 3\tan x + 1 = 0 \) is in the form \(ax^2+bx+c=0\). Here, instead of 'x', we have '\tan x'. Solving for '\tan x' gives us the equation in quadratic form: \(\tan x = \frac{3 ± \sqrt{(-3)^2 -4(1)(1)}}{2*(1)}\).

Step 2: Use the quadratic formula

Formulate the quadratic formula for this equation, \(\tan x = \frac{3 ± \sqrt{9 -4}}{2}\). That is \(\tan x = \frac{3 ± \sqrt{5}}{2}\). So, the two possible values are \(\tan x =\frac{3 + \sqrt{5}}{2} \) and \(\tan x =\frac{3 - \sqrt{5}}{2}\). This implies that \(x = \tan^{-1}(\frac{3 + \sqrt{5}}{2})\) and \(x = \tan^{-1}(\frac{3 - \sqrt{5}}{2})\).

Step 3: Solve for x

Calculate for 'x' using the inverse tangent function and a calculator for the precise decimal answer. Make sure that the solutions fall within the given interval [0, 2π).

Step 4: Verify the solutions

Substitute the solutions obtained in the original equation to ensure that the equation holds true.

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation that can be represented in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The solutions to a quadratic equation are found using the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Notably, the term under the square root in the formula is known as the discriminant \(b^2 - 4ac\). It helps determine the nature of the roots:

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there is one real solution.
• If it is negative, the solutions are complex.

In the case of our problem, we're dealing with an equation where the variable is \(\tan x\), but the process remains the same. By setting \(\tan x\) like 'x', we use the quadratic formula to derive potential values of \(\tan x\). Exploring these fundamentals can greatly enhance understanding and application in solving more advanced algebraic and trigonometric problems.

Tangent Function

The tangent function, denoted as \(\tan(x)\), is a primary trigonometric function. It is defined as the ratio of the sine and cosine functions:

• \(\tan x = \frac{\sin x}{\cos x}\).

Tangent is periodic with a period of \(\pi\), which means it repeats its values every \(\pi\) radians. This function can take any real number values and is positive where it appears in the first and third quadrants of the unit circle.

• Tangent is undefined where \(\cos x = 0\).

When solving trigonometric equations involving \(\tan x\), it is vital to consider this periodicity and the quadrant to which the angle corresponds. By using these principles, one can find all possible solutions that fit within a specified interval.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine the angle when given the function's value. For the tangent function, we use \(\tan^{-1}(x)\), commonly referred to as \(\arctan(x)\). The inverse tangent will provide an angle whose tangent is \(x\).

• On a calculator, \(\arctan\) often gives results in radians and restricts angles to the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

However, when solving trigonometric equations, be mindful that the tangent function is periodic. Thus, we need to find all possible solutions within the desired interval. For our given problem, solutions need to fall within \([0, 2\pi)\). The use of inverse trigonometric functions ensures that we can calculate these solutions precisely using a calculator to four decimal places. Understanding this function and its properties aids in distinguishing between potential solutions and selecting the correct ones.

Problem-solving Steps in Trigonometry

Solving trigonometric equations can be simplified by following a structured approach:

• Identify the function and equation type: Recognize that the equation is trigonometric and may mimic algebraic forms like quadratic or linear.
• Solve algebraically: Apply any necessary algebraic techniques, such as factoring or using the quadratic formula, to reveal possible trigonometric solutions.
• Use inverse trigonometric functions: Determine potential angle solutions using inverse functions like \(\arctan\).
• Consider periodicity: Make use of the periodic nature of trigonometric functions to identify all solutions over a specified interval.
• Verify solutions: Substitute back into the original equation to confirm accuracy within constraints.

By implementing these problem-solving steps, complex trigonometric equations, such as the one given in the exercise, become much more approachable, guiding students to accurate solutions.