Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right) .\) Give solutions to the nearest tenth as appropriate. $$8 \cos ^{2} \theta+18 \cos \theta=5$$

Step-by-Step Solution

Answer

The solutions for \( \theta \) are approximately \( 75.5^\circ \) and \( 284.5^\circ \).

Step 1: Rearrange the Equation

We begin by rearranging the equation: \[ 8 \cos^2 \theta + 18 \cos \theta - 5=0 \]This is a quadratic equation in terms of \( \cos \theta \).

Step 2: Identify Substitution

Substitute \( x = \cos \theta \). This gives us:\[ 8x^2 + 18x - 5 = 0 \]

Step 3: Use the Quadratic Formula

Use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 8 \), \( b = 18 \), and \( c = -5 \). Calculate the discriminant:\[ b^2 - 4ac = 18^2 - 4 \times 8 \times (-5) = 324 + 160 = 484 \]Now apply the quadratic formula:\[ x = \frac{-18 \pm \sqrt{484}}{16} \]

Step 4: Calculate the Roots

Calculate the solutions for \( x \):\[ x = \frac{-18 + 22}{16} = \frac{4}{16} = 0.25 \]\[ x = \frac{-18 - 22}{16} = \frac{-40}{16} = -2.5 \]Since \(-2.5\) is not a valid cosine value (cosine values range from \(-1\) to \(1\)), we discard it. Thus, \( \cos \theta = 0.25 \).

Step 5: Find Angles for Cosine Value

Solve \( \cos \theta = 0.25 \) over the interval \([0^\circ, 360^\circ)\). This gives two possible angles \( \theta \) using a calculator:\[ \theta = \cos^{-1}(0.25) \approx 75.5^\circ \]\( \cos \theta = 0.25 \) is also true for the angle in the fourth quadrant:\[ \theta = 360^\circ - 75.5^\circ = 284.5^\circ \]

Step 6: Verify Answers

Check that both \( 75.5^\circ \) and \( 284.5^\circ \) work for the original equation. Both are within the valid interval \([0^\circ, 360^\circ)\), thus they are valid solutions.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra that expresses a polynomial of degree two. This means that the highest power of the variable is squared, or in mathematical terms, it has the form:\[a x^2 + b x + c = 0\]In our specific problem, we recognize that the arrangement \[8 \cos^2 \theta + 18 \cos \theta - 5 = 0\]is a quadratic equation where the unknown variable is the cosine of an angle, \(\cos \theta\). Solving quadratic equations typically involves techniques like factoring (if possible), completing the square, or using the quadratic formula. Given the coefficients, here the quadratic formula is the most straightforward solution method. This reliable formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\]where \(a\), \(b\), and \(c\) are constants. In essence, solving quadratic equations involves finding the values of \(x\) that satisfy the equation.

Cosine Function

The cosine function is a fundamental trigonometric function that relates the measure of an angle in a right triangle to the lengths of the sides. In a unit circle, the cosine of an angle \(\theta\) is the x-coordinate of the point where the terminal side of the angle intersects the circle. The cosine function is periodic with a period of \(360^\circ\) (or \(2\pi\) radians).Key properties of the cosine function include:

• It is symmetric about the y-axis, meaning it is an even function.
• Values range from \(-1\) to \(1\).
• When solving the equation \(\cos \theta = c\), potential solutions are angles where the x-coordinate corresponds to \(c\).

For example, if \(\cos \theta = 0.25\), we find the angle(s) \(\theta\) that satisfy this condition within a specified interval by referring to the unit circle or using inverse trigonometric functions like \(\cos^{-1}\).

Interval Solutions

Interval solutions in trigonometry refer to finding all possible angles that satisfy a given equation within a specific range. In our exercise, the goal is to find angles \(\theta\) where \(0^\circ \leq \theta < 360^\circ\). Such intervals ensure we find solutions within one full rotation around a circle, which also represents a typical cycle of trigonometric functions.When solving \(\cos \theta = 0.25\), it's essential to identify all such angles across the defined interval. The primary angle, found through \(\cos^{-1}(0.25)\), gives one solution. However, due to the periodic nature of cosine, another solution exists in the complementary quadrant (the fourth quadrant in this scenario). Therefore, a complete set of solutions includes these alternate angles as well, reaffirming that trigonometric functions can have more than one solution in a given cycle.

Unit Circle

The unit circle is a crucial concept in understanding trigonometric functions. It is a circle with a radius of one, centered at the origin of the coordinate system. The unique property of the unit circle is that it allows easy derivation of the sine and cosine values for any angle measured in radians or degrees.On the unit circle:

• The x-coordinate gives the cosine of the angle.
• The y-coordinate gives the sine of the angle.
• As the angle increases from \(0^\circ\) to \(360^\circ\), these coordinates trace the circumference of the circle.

Given its symmetrical nature and uniform radius, the unit circle is an excellent tool for visualizing why trigonometric functions are periodic and how they repeat values. Understanding its geometry helps in solving problems like finding interval solutions for cosine equations, as these solutions often rely on determining the angles that correspond to specific x-coordinates around the circle.