Problem 56 - A Graphical Approach to Precalculus with Limits

Question

Give solutions over the interval \([0,2 \pi)\) as approximations to the nearest hundredth when exact values cannot be determined. You may need to use the quadratic formula. Give approximate answers in Exercises \(59-64\) to the nearest tenth of a degree over the interval \(\left[0^{\circ}, 360^{\circ}\right)\) $$3 \cot ^{2} x-3 \cot x=1$$

Step-by-Step Solution

Answer

The approximate solutions are \(x \approx 0.51\) and \(x \approx 1.67\, \text{radians}\).

Step 1: Rewrite the Equation

Start with the given equation: \(3 \cot^2 x - 3 \cot x = 1\). We can express this equation in a standard quadratic form. Move all the terms to one side to set the equation to zero:\[3 \cot^2 x - 3 \cot x - 1 = 0\].

Step 2: Substitute for Simplicity

Let \(y = \cot x\). Substitute \(y\) into the equation: \(3y^2 - 3y - 1 = 0\). This makes solving for \(y\) more straightforward.

Step 3: Apply the Quadratic Formula

The quadratic formula is \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=3\), \(b=-3\), and \(c=-1\). Plug these values into the formula:\[y = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \times 3 \times (-1)}}{2 \times 3} = \frac{3 \pm \sqrt{9 + 12}}{6} = \frac{3 \pm \sqrt{21}}{6}\].

Step 4: Calculate y Values

Calculate the approximate values for \(y\):1. \(y_1 = \frac{3 + \sqrt{21}}{6}\) which is approximately \(1.76\).2. \(y_2 = \frac{3 - \sqrt{21}}{6}\) which is approximately \(-0.10\).

Step 5: Solve for x Using Inverse Cotangent

Now, convert \(y\) back to \(x\) using the inverse cotangent:1. For \(y_1 = 1.76\), \(x = \cot^{-1}(1.76)\). This results in \(x \approx 29.18^{\circ}\).2. For \(y_2 = -0.10\), \(x = \cot^{-1}(-0.10)\). This results in \(x \approx 95.71^{\circ}\).

Step 6: Convert Degrees to Radians

Convert the degree solutions back to radians since the initial range is \([0, 2\pi)\):1. For \(29.18^{\circ}\), convert to radians: \(x \approx 0.51\, \text{radians}\).2. For \(95.71^{\circ}\), convert to radians: \(x \approx 1.67\, \text{radians}\).

Step 7: Verify Solutions

Check the approximate solutions by plugging them back into the original equation to ensure they satisfy it.- Check for both \(x \approx 0.51\) and \(x \approx 1.67\) whether \(3 \cot^2 x - 3 \cot x \approx 1\). Both should be close, verifying the solutions.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). When an equation cannot be easily factored, the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), helps find the solutions efficiently.

Here's a quick breakdown of how to use it:

• Identify the coefficients \(a\), \(b\), and \(c\) in the equation.
• Substitute these values into the formula.
• Simplify under the square root to solve for the two possible values of \(x\).

Take care to evaluate the discriminant, \(b^2 - 4ac\). This determines the nature of the roots:

• If positive, you have two distinct real solutions.
• If zero, there is one real solution.
• If negative, you won't have real solutions (only complex ones).

In our exercise, substituting \(a=3\), \(b=-3\), and \(c=-1\) into the formula helps us find approximate solutions for \(y\), representing \(\cot x\). Now, you can proceed to find \(x\) using these \(y\) values.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find the angle in a right triangle when a trigonometric ratio is known. The inverse of the cotangent, \(\cot^{-1}(x)\), is particularly useful here. Given a \(y\) value as a result of using the quadratic formula, inverse functions help us solve for the angle \(x\).

Here’s how you apply it:

• For \(y_1 = 1.76\), calculate \(x = \cot^{-1}(1.76)\), which results in \(x \approx 29.18^{\circ}\).
• For \(y_2 = -0.10\), calculate \(x = \cot^{-1}(-0.10)\), resulting in \(x \approx 95.71^{\circ}\).

Remember that the range of \(\cot^{-1}\) generally needs consideration, as inverse functions can yield multiple angles that satisfy a trigonometric equation. Make sure the solution angles are within the specified range before converting them to radians.

Degree to Radian Conversion

Converting degrees to radians is a crucial step, especially when dealing with trigonometric equations that require solutions in radian measure over intervals like \([0, 2\pi)\). This conversion uses the basic relationship that \(180^{\circ} = \pi\) radians.

To convert an angle from degrees to radians, use the formula:

\( \text{radians} = \left( \frac{\text{degrees} \times \pi}{180} \right) \)

Let's apply this to our computed angles:

• For \(29.18^{\circ}\), the conversion is \( x \approx 0.51\, \text{radians} \).
• For \(95.71^{\circ}\), the conversion results in \( x \approx 1.67\, \text{radians} \).

By converting degrees to radians, you can express the solutions of trigonometric equations appropriately within the specified radian interval. This not only aligns with the problem's requirements but also provides a uniform way to interpret angles across different mathematical applications.