Problem 64 - 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 \(\left[0^{\circ}, 360^{\circ}\right)\) $$\tan ^{2} \theta+2 \sec \theta-3=0$$

Step-by-Step Solution

Answer

The solutions are \(\theta \approx 51.7^{\circ}\) and \(\theta \approx 308.3^{\circ}\).

Step 1: Convert Trigonometric Equation Into a Solvable Form

Given \(\tan ^{2} \theta+2 \sec \theta-3=0\), let's rewrite it in terms of a common trigonometric function. Recall that \(\tan^2 \theta = \sec^2 \theta - 1\). Therefore, replace \(\tan^2 \theta\) and rewrite the equation as \(\sec^2 \theta - 1 + 2 \sec \theta - 3 = 0\). Simplify to get \(\sec^2 \theta + 2 \sec \theta - 4 = 0\).

Step 2: Apply Quadratic Formula

The equation \(\sec^2 \theta + 2 \sec \theta - 4 = 0\) is in quadratic form \(ax^2 + bx + c = 0\) where \(a = 1\), \(b = 2\), and \(c = -4\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(x = \sec \theta\). Calculate: \(b^2 - 4ac = 2^2 - 4(1)(-4) = 4 + 16 = 20\).

Step 3: Find Solutions Using the Quadratic Formula

Substitute in the quadratic formula: \(\sec \theta = \frac{-2 \pm \sqrt{20}}{2}\). Simplifying: \(\sec \theta = \frac{-2 \pm 2\sqrt{5}}{2}\), which gives us \(\sec \theta = -1 \pm \sqrt{5}\).

Step 4: Solve for Angular Solutions

Since \(\sec \theta = \frac{1}{\cos \theta}\), we substitute to find \(\cos \theta = -\frac{1}{1 + \sqrt{5}}\) and \(\cos \theta = \frac{1}{\sqrt{5} - 1}\). The first value is approximately invalid since \(-1\) is not in the range of \(\cos \theta\), but we calculate the second value to get actual angles. Convert \(\cos \theta = \frac{1}{\sqrt{5} - 1}\) into decimal form (using a calculator) to approximate \(\cos \theta \approx 0.618\).

Step 5: Calculate Solutions in radians over \([0, 2\pi)\)

Using a calculator, determine \(\theta\) such that \(\cos \theta \approx 0.618\). The primary angle is \(\theta \approx 0.902\) radians. Also consider \(\theta = 2 \pi - 0.902\) radians to account for the symmetry in the unit circle. Calculate \(2 \pi - 0.902 \approx 5.381\).

Step 6: Convert Radial Solutions to Degrees

To convert \(\theta\) from radians to degrees, multiply by \(\frac{180}{\pi}\). So, for \(\theta = 0.902\), convert to degrees: \(0.902 \times \frac{180}{\pi} \approx 51.67\) degrees. Similarly, \(5.381 \times \frac{180}{\pi} \approx 308.33\) degrees.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool to solve quadratic equations, equations that can be expressed in the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown value. The formula is given by:

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

This formula helps find the values of \(x\) that make the equation true by utilizing the coefficients \(a\), \(b\), and \(c\). It is derived from completing the square method and works by calculating two potential solutions for \(x\) by adding or subtracting the square root term.
In the trigonometric equation provided in the exercise, \( \sec \theta \) is treated as \(x\) when using the quadratic formula. This allows solving for possible angles \(\theta\) that fit the equation.

Radians to Degrees Conversion

Radians and degrees are two different units for measuring angles. Often in trigonometry, angles are provided in radians, but for many practical applications, it can be easier to understand these angles if they are converted into degrees. This is done using the conversion factor:

• \(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\).

To convert radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\).
For example, as shown in the step-by-step solution, angles like \(0.902\) radians are converted into degrees by calculating \(0.902 \times \frac{180}{\pi} \approx 51.67\) degrees. This can help in understanding the position of the angle in a more familiar 360-degree system.

Unit Circle

The unit circle is a fundamental tool in trigonometry, representing all possible angle measures in standard position (where the vertex is at the origin and one ray is on the positive x-axis). The circle is centered at the origin with a radius of 1.
Key properties of the unit circle include:

• The x-coordinate corresponds to \(\cos \theta\),
• The y-coordinate corresponds to \(\sin \theta\),
• All angles can be measured clockwise or counterclockwise from the positive x-axis.

This circle helps in understanding angle measures in radians and degrees, as well as the periodic nature of sine, cosine, and tangent functions.
The unit circle also helps visualize solutions for trigonometric equations like the one in the exercise, facilitating the understanding of multiple-angle solutions such as \(\theta = 0.902\) and \(\theta = 5.381\), which are correspondingly spaced elsewhere on the circle.

Inverse Trigonometric Functions

Inverse trigonometric functions help find the angle that corresponds to a given trigonometric value. These functions are essentially the "reverse" processes of the standard trig functions. They include:

• \(\sin^{-1}(x)\) or arcsin, which identifies an angle with a given sine value
• \(\cos^{-1}(x)\) or arccos, which identifies an angle with a given cosine value
• \(\tan^{-1}(x)\) or arctan, which identifies an angle with a given tangent value

Through these functions, we can find possible angles that correspond to specific trigonometric values modulo the periodicity of trig functions.
When solving equations like \(\sec \theta = \frac{1}{\cos \theta} = \text{value}\), using inverse cosine functions can help find the associated angle \(\theta\) on the unit circle, giving precise understanding of the problem at hand.