Problem 70 - 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. $$3 \sin ^{2} \theta-\sin \theta=2$$

Step-by-Step Solution

Answer

The solutions are \( \frac{\pi}{2}, 3.8713, 5.5535 \) radians or 90°, 221.8°, and 318.2° in degrees.

Step 1: Substitute for \( ext{sin} \theta \)

First, begin by letting \( x = \sin \theta \). This substitution transforms the trigonometric equation into a quadratic form: \( 3x^2 - x = 2 \).

Step 2: Rearrange into a standard form equation

Rearrange the equation to form a standard quadratic equation: \( 3x^2 - x - 2 = 0 \).

Step 3: Use the quadratic formula to solve

The quadratic formula is given by \( x = \frac{-b \pm \, \sqrt{b^2-4ac}}{2a} \). Here, \( a = 3 \), \( b = -1 \), and \( c = -2 \). Substitute these values into the formula:\[x = \frac{-(-1) \pm \, \sqrt{(-1)^2 - 4 \times 3 \times (-2)}}{2 \times 3}\]Calculate the discriminant \( (-1)^2 - 4 \times 3 \times (-2) = 1 + 24 = 25 \), giving:\[x = \frac{1 \pm \, \sqrt{25}}{6}\]

Step 4: Evaluate solutions from the quadratic formula

Solve the equation for \( x \):\[x = \frac{1 + 5}{6} = 1\;\text{or}\; x = \frac{1 - 5}{6} = -\frac{2}{3}\]Since \( x = \sin \theta \), and \( \sin \theta \) cannot be greater than 1, the solution \( x = 1 \) corresponds to \( \theta = \frac{\pi}{2} \) radians or 90° in degrees. For \( x = -\frac{2}{3} \), continue to the next step.

Step 5: Solve for \( \theta \) when \( \sin \theta = -\frac{2}{3} \)

Find \( \theta \) where \( \sin \theta = -\frac{2}{3} \). The reference angle \( \alpha \) corresponding to \( \sin^{-1}\left( \frac{2}{3} \right) \) is approximately 0.7297 radians or 41.8°. Since \( \sin \theta \) is negative in the third and fourth quadrants, add \( \alpha \) to \( \pi \) (or 180°) and \( 2\pi - \alpha \) (or 360° - \( \alpha \)) for the exact values:- Third quadrant: \( \theta = \pi + 0.7297 \approx 3.8713 \) radians or \( 180 + 41.8 = 221.8° \).- Fourth quadrant: \( \theta = 2\pi - 0.7297 \approx 5.5535 \) radians or \( 360 - 41.8 = 318.2° \).

Step 6: Combine all solutions

Combine the solutions to present the nonnegative least angle measures: \( \theta = \frac{\pi}{2}, 3.8713, 5.5535 \) radians or 90°, 221.8°, and 318.2° in degrees.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool in solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula allows us to find the roots of the equation, or in simpler terms, the values of \( x \) that make the equation true. The formula is expressed as:

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

To use the quadratic formula, it's necessary to identify the coefficients: \( a \), \( b \), and \( c \). In the equation \( 3x^2 - x - 2 = 0 \), \( a = 3 \), \( b = -1 \), and \( c = -2 \). By substituting these values into the formula, you can calculate the possible solutions for \( x \) or in our context for \( \sin \theta \). Remember that the term under the square root, \( b^2 - 4ac \), is known as the discriminant and it determines the nature of the roots (real or complex). Here, a positive discriminant (25) indicates two distinct real solutions.

Sine Function

The sine function, represented as \( \sin \theta \), is one of the primary trigonometric functions. It is useful for relating the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. For our exercise, \( \sin \theta \) appears in the transformed quadratic equation we solved.

• Mathematically, \( \sin \theta = \frac{opposite}{hypotenuse} \)

In the equation \( 3 \sin^2 \theta - \sin \theta = 2 \), we made a substitution with \( x = \sin \theta \), which turned it into a solvable quadratic equation. The solutions to this provided the possible values of \( \sin \theta \) which were \( 1 \) and \(-\frac{2}{3}\). It's important to know that the range of \( \sin \theta \) is between -1 and 1. Thus, valid solutions lie within this range. In this exercise, \( \theta \) provided unique angle solutions for these sine values.

Radians and Degrees Conversion

Understanding radians and degrees is essential in trigonometry, as angles can be expressed in either unit. Radians measure angles based on the arc length of a circle, whereas degrees are based on dividing a circle into 360 equal parts. To convert between radians and degrees, use the following conversions:

• To convert from radians to degrees, use \( degrees = radians \times \frac{180}{\pi} \)
• To convert from degrees to radians, use \( radians = degrees \times \frac{\pi}{180} \)

In this exercise, we found angle solutions in both radians and degrees. For instance, \( \frac{\pi}{2} \) radians is equivalent to 90 degrees. Converting these accurately ensures that the answer is represented in the desired unit, allowing clearer communication and understanding.

Angle Measure

In trigonometry, angle measures indicate the size of an angle, which can be expressed in either radians or degrees. Importantly, choosing the right representation can help make problem-solving straightforward. The sine function generates different angles depending on the quadrant in the unit circle. For a sine value of \(-\frac{2}{3}\), we found different angle measures in the third and fourth quadrants as follows:

• Third Quadrant: Adding the reference angle to 180° gives us \(\theta = 221.8^{\circ}\).
• Fourth Quadrant: Subtracting the reference angle from 360° results in \(\theta = 318.2^{\circ}\).

Understanding which angles correspond to specific sine values is crucial, as it ensures the correct nonnegative least angle measures are used. Each angle measure gives clues to its position on the unit circle, facilitating deeper insights and the solution of trigonometric equations like \( 3\sin^{2}\theta - \sin\theta = 2 \).