Problem 69 - A Graphical Approach to Precalculus with Limits

Question

Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$6 \sin ^{2} \theta+\sin \theta=1$$

Step-by-Step Solution

Answer

The solutions are \(\theta \approx 0.3398\) radians (19.5°) and \(\theta \approx 5.7596\) radians (330°).

Step 1: Set Equation to Standard Form

Given the equation \(6 \sin^2 \theta + \sin \theta = 1\). Start by bringing all terms to one side of the equation: \(6 \sin^2 \theta + \sin \theta - 1 = 0\).

Step 2: Substitute Variable for Simplification

Substitute \(u = \sin \theta\). The equation becomes \(6u^2 + u - 1 = 0\).

Step 3: Use the Quadratic Formula

Use the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=6\), \(b=1\), and \(c=-1\). Calculate the discriminant: \(b^2 - 4ac = 1 + 24 = 25\). The roots are \(u = \frac{-1 \pm 5}{12}\).

Step 4: Calculate Possible Values of \(u\)

Calculate \(u_1 = \frac{-1 + 5}{12} = \frac{4}{12} = \frac{1}{3}\) and \(u_2 = \frac{-1 - 5}{12} = \frac{-6}{12} = -\frac{1}{2}\).

Step 5: Solve for \(\theta\) When \(u = \sin \theta = \frac{1}{3}\)

\(\sin \theta = \frac{1}{3}\) implies \(\theta = \sin^{-1}\left(\frac{1}{3}\right)\). Calculating, \(\theta \approx 0.3398\) radians or \(\theta \approx 19.5\) degrees.

Step 6: Solve for \(\theta\) When \(u = \sin \theta = -\frac{1}{2}\)

\(\sin \theta = -\frac{1}{2}\) implies \(\theta = \sin^{-1}\left(-\frac{1}{2}\right)\). Calculating, \(\theta = -\frac{\pi}{6}\) or \(\theta = 11\frac{\pi}{6}\) in radians, and \(\theta \approx 330\) degrees.

Step 7: Conclusion with Least Nonnegative Angles

The least nonnegative angle measures are \(\theta \approx 0.3398\) radians (or \(\theta \approx 19.5\) degrees) and \(\theta \approx 5.7596\) radians (or \(\theta \approx 330\) degrees).

Key Concepts

Radians and Degrees

Trigonometry often requires converting between radians and degrees. Each unit represents a different way to measure angles.
Radians are based on the radius of a circle. A full circle is equal to \(2\pi\) radians, similar to 360 degrees in degrees measurement.
To convert from degrees to radians, multiply by \(\frac{\pi}{180}\). For example, 180 degrees equals \(\pi\) radians.

• When rounded to four decimal places, small differences in values can occur. This is crucial for precision in mathematics.
• Rounding degrees to the nearest tenth involves looking at the digit in the hundredths place to make this decision.

Understanding these conversions is critical in solving trigonometric equations, as some solutions require one format over the other.

Quadratic Formula

The Quadratic Formula is a powerful tool for solving quadratic equations, which have the standard form \(ax^2 + bx + c = 0\).
This formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Using this formula allows you to find the roots or solutions of the equation.
Important steps include:

• Identify \(a\), \(b\), and \(c\), the coefficients of the terms.
• Calculate the discriminant \(b^2 - 4ac\). This determines the nature of the roots; real and distinct when the discriminant is positive.
• Calculate the solutions using the formula.

In the context of trigonometry, replacing \(x\) with a trigonometric function like \(\sin \theta\) transforms the process to solve for angles, as seen in the provided equation.

Sine Function

The Sine Function is one of the basic trigonometric functions, representing the ratio of the opposite side to the hypotenuse in a right triangle.
This function is periodic, repeating every \(2\pi\) radians (or 360 degrees).
It has a range of \([-1, 1]\), making it ideal for many applications like solving equations.

• To find the angle \(\theta\) when given \(\sin \theta = y\), use the inverse sine function \(\sin^{-1}(y)\).
• Keep the range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\) in mind for inverse sine calculations to ensure correct angle measures.

In equations, the sine function helps determine possible angle solutions, which are verified by plugging back into the expression to meet given conditions, such as being nonnegative.

Nonnegative Angle Measures

For solving trigonometric equations, we often seek nonnegative angle measures. This ensures that the solutions make sense in the context of a full circle, or 360 degrees.
Nonnegative angles range from 0 to 360 degrees (or 0 to \(2\pi\) radians).

• To express solutions as nonnegative angles, always add \(360\) degrees (or \(2\pi\) radians) to any negative angle until you reach a corresponding positive equivalent.
• Check variations of the inverse function solutions to ensure you're covering all possible valid angles within one rotation of a circle.

Providing solutions as the least possible nonnegative angles is a common requirement to avoid ambiguities and ensure a standard form for answer representation.