Problem 42 - Precalculus Mathematics for Calculus

Question

Solve the given equation. $$2 \sin ^{2} \theta-\sin \theta-1=0$$

Step-by-Step Solution

Answer

The solutions are \(\theta = \frac{7\pi}{6}, \frac{11\pi}{6},\) and \(\frac{\pi}{2}\).

Step 1: Identify the Trigonometric Identity

Notice that the equation \(2 \sin^2 \theta - \sin \theta - 1 = 0\) resembles a quadratic equation, where \(\sin \theta\) is like the variable \(x\). This suggests we can solve it using the methods applied to quadratic equations.

Step 2: Substitute and Simplify

Let \( u = \sin \theta \). The equation becomes \(2u^2 - u - 1 = 0\). Now, it is a standard quadratic equation, which we will solve for \(u\) next.

Step 3: Factor the Quadratic Equation

We need two numbers that multiply to \(-2\) (from \(2 \times -1\)) and add to \(-1\). These numbers are \(-2\) and \(1\). Factor the equation as \((2u + 1)(u - 1) = 0\).

Step 4: Solve Each Factor

Set each factor equal to zero: For \(2u + 1 = 0\), solve to get \(u = -\frac{1}{2}\).For \(u - 1 = 0\), solve to get \(u = 1\).

Step 5: Substitute Back to Trigonometric Function

Recall that \(u = \sin \theta\), so we have \(\sin \theta = -\frac{1}{2}\) and \(\sin \theta = 1\). Solve for \(\theta\) in each case.

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

\(\sin \theta = -\frac{1}{2}\) gives solutions at \(\theta = \frac{7\pi}{6}\) and \(\theta = \frac{11\pi}{6}\) in the interval \([0, 2\pi)\).

Step 7: Solve for \(\theta\) Using \(\sin \theta = 1\)

\(\sin \theta = 1\) gives a solution at \(\theta = \frac{\pi}{2}\).

Step 8: Final Solution

Combine all the solutions obtained for \(\theta\): \(\theta = \frac{7\pi}{6}, \frac{11\pi}{6}, \) and \(\frac{\pi}{2}\) in the interval \([0, 2\pi)\).

Key Concepts

Quadratic Equations in Trigonometry

When solving equations in trigonometry that have a quadratic form, it's useful to think of trigonometric functions like sine or cosine as variables. Just as you would solve a standard quadratic equation like \( ax^2 + bx + c = 0 \), you can approach a trigonometric equation such as \( 2\sin^2\theta - \sin\theta - 1 = 0 \) in a similar manner.

The key steps involve:

• Identifying the quadratic nature of the equation. In this case, \( \sin \theta \) is akin to the variable \( x \) in a typical quadratic equation.
• Substituting \( \sin \theta \) with a generic variable like \( u \), transforming it into a form \( 2u^2 - u - 1 = 0 \).

This transformation allows you to use factoring, the quadratic formula, or other algebraic methods to solve the equation.

Ultimately, once you find the solutions for your substituted variable, remember to substitute back your trigonometric function to get the answers in the form of angles, which is your final goal when working with trigonometric equations.

Trigonometric Identities

Trigonometric identities serve as useful tools that allow for transformation and simplification of complex trigonometric expressions. In the context of solving trigonometric equations, knowing these identities can provide shortcuts and deepen the understanding of how to approach each problem component.

Some common identities you might encounter include:

• Pythagorean identity: \( \sin^2\theta + \cos^2\theta = 1 \)
• Double angle identities, such as \( \sin(2\theta) = 2\sin\theta\cos\theta \)
• Addition and subtraction formulas, like \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \)

For our equation \( 2 \sin^2\theta - \sin\theta - 1 = 0 \), recognizing it as resembling the structure of a quadratic equation is an extension of understanding basic trigonometric properties. While trigonometric identities aren't directly used in factoring this equation, they are vital for simplifying equations or exchanging one trigonometric function for another when trying to solve complex problems.

Angle Solutions in Radians

In trigonometry, solutions to equations are typically given in terms of angles, and these angles are often expressed in radians. A radian is a measure of an angle that is directly tied to the arc length of a circle. Understanding radian measure is crucial for solving and interpreting solutions to trigonometric equations.

Here's what you need to remember:

• The full circle in radians is \( 2\pi \).
• Common angles include \( \pi/2 \) (90 degrees), \( \pi \) (180 degrees), \( 3\pi/2 \) (270 degrees), and \( 2\pi \) (360 degrees).
• For the equation, \( \sin \theta = -1/2 \), the angle solutions are \( \theta = 7\pi/6 \) and \( 11\pi/6 \).
• When \( \sin \theta = 1 \), the angle solution is \( \pi/2 \).

Understanding these angles within the interval \([0, 2\pi)\) is essential, as it encompasses all solutions without repetition. Using radians gives a more natural and direct connection when dealing with calculus or more advanced mathematics.