Problem 16 - Algebra 2 and Trigonometry

Question

Find, to the nearest hundredth of a radian, all values of \(\theta\) in the interval \(0 \leq \theta<2 \pi\) for which \(\frac{\cos \theta}{3}=\frac{1}{3 \cos \theta+1}\)

Step-by-Step Solution

Answer

\( \theta = 0 \) radians

Step 1: Set the Equation Equal

We start with the given equation: \( \frac{\cos \theta}{3} = \frac{1}{3 \cos \theta + 1} \). Our goal is to solve for \( \theta \) within the specified interval \( 0 \leq \theta < 2 \pi \). Since both sides are fractions, we can start by equating the numerators against the condition that cross-multiplying should not make the denominator zero.

Step 2: Cross-Multiply

Cross-multiply to eliminate the fractions: \( \cos \theta (3 \cos \theta + 1) = 3 \times 1 \). This leads to the equation: \( 3 \cos^2 \theta + \cos \theta = 3 \).

Step 3: Rearrange as a Quadratic Equation

Rearrange the equation to form a quadratic equation: \( 3 \cos^2 \theta + \cos \theta - 3 = 0 \). This is a standard quadratic in terms of \( \cos \theta \).

Step 4: Solve the Quadratic Equation

Use the quadratic formula to solve for \( \cos \theta \): \[ \cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 3 \), \( b = 1 \), and \( c = -3 \). Substitute these values into the formula: \[ \cos \theta = \frac{-1 \pm \sqrt{1 + 36}}{6} = \frac{-1 \pm 7}{6} \].This provides two solutions for \( \cos \theta \): \( \cos \theta = 1 \) and \( \cos \theta = -\frac{4}{3} \). Since the cosine range is only between \(-1\) and \(1\), the value \( \cos \theta = -\frac{4}{3} \) is extraneous.

Step 5: Find the Angle for Valid Cosine Value

For the valid \( \cos \theta = 1 \), \( \theta \) must be \( 0 \) radians because cosine is 1 at \( \theta = 0 \) within the range \( 0 \leq \theta < 2\pi \).

Step 6: Verify the Solution in Given Range

Ensure \( \theta = 0 \) is in the given range. Since it satisfies the equation and is within \( 0 \leq \theta < 2\pi \), it is a valid solution.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function, commonly involved in problems concerning angles and triangles. It identifies how an angle in a right triangle relates to the ratio of the length of the adjacent side to the hypotenuse. Mathematically, for an acute angle \( \theta \), it is expressed as:

• \( \cos \theta = \frac{{\text{adjacent}}}{{\text{hypotenuse}}} \)

The cosine function is defined and continuous for all real numbers, but it exhibits a periodic behavior, repeating every \(2\pi\) radians. This property makes it particularly useful in solving trigonometric equations over specified intervals.
Another essential feature of the cosine function is its range, which is limited to values between \(-1\) and \(1\). Therefore, when solving equations involving \( \cos \theta \), it is necessary to consider only those solutions within this interval. For instance, if a solution like \( \cos \theta = -\frac{4}{3} \) arises, it must be discarded because it lies outside the permissible range.
In the original problem, determining \( \theta \) when \( \cos \theta = 1 \) leads to a straightforward solution: \( \theta = 0 \), as this is where cosine reaches its maximum value within the specified interval \(0 \leq \theta < 2\pi \).

Quadratic Equations

Quadratic equations are polynomial equations of degree two and are represented in the standard form as:

• \( ax^2 + bx + c = 0 \)

In the context of trigonometric equations, such as \( 3 \cos^2 \theta + \cos \theta - 3 = 0 \), they often surface after manipulating the original equation to isolate trigonometric functions. The key to solving these is the quadratic formula:

• \[ \cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In the given exercise, substituting \( a = 3 \), \( b = 1 \), and \( c = -3 \) into the quadratic formula yields potential solutions for \( \cos \theta \). The subsequent analysis filters out extraneous values that do not fall within the cosine's valid range.
These types of problems require proficiency in algebraic manipulations and understanding the properties of cosine to determine which solutions are legitimate.
This approach outlines a systematic route to explore different values of \( \theta \) that satisfy the quadratic equation while adhering to the conditions of trigonometric functions.

Radian Measure

Radian measure is a way to express angles based on the radius of a circle. It provides a natural measure for angles as opposed to degrees. Understanding radian measure is crucial because many trigonometric calculations and equations deal primarily in radians, especially in higher mathematics.
One radian corresponds to the angle created when the arc length is equal to the radius of the circle. In a full circle, there are \(2\pi\) radians, equivalent to 360 degrees. Thus, \(\pi\) radians equal 180 degrees, making conversions between degrees and radians quite intuitive.
In the current exercise, finding \( \theta \) to the nearest hundredth of a radian involves ensuring that solutions for \( \theta \), such as \(0\) radians, adhere to the radian interval \(0 \leq \theta < 2\pi \). Understanding and working within this framework are necessary to assess the correctness of solutions and guarantee their appropriateness within the problem's specified range.
This measure helps to ensure that solutions are precise and universally interpretable, a critical aspect when solutions must accommodate multiple mathematical contexts.