Problem 65 - Algebra and Trigonometry

Question

Use an identity to solve each equation on the interval \([0,2 \pi)\) $$ \sin ^{2} x-2 \cos x-2=0 $$

Step-by-Step Solution

Answer

The equation \( \sin^2{x} - 2\cos{x} - 2 = 0 \) has no solutions in the interval [0, 2π).

Step 1: Convert sine-square term into cosine-square term

Using the Pythagorean trigonometric identity \( \sin^2{x} = 1 - \cos^2{x} \), replace \( \sin^2{x} \) in the equation with \( 1 - \cos^2{x} \). This gives the following equation: \( 1 - \cos^2{x} - 2\cos{x} - 2 = 0 \).

Step 2: Simplify the equation

By rearranging the terms in the equation, we get: \( - \cos^2{x} - 2 \cos{x} + 1 = 0 \) . Now, rearrange the equation to fit the standard quadratic form \( ax^2 + bx + c = 0 \). To do this, the equation becomes: \( \cos^2{x} + 2\cos{x} - 1 = 0 \).

Step 3: Solve for cosine

Now, treat this equation as you would any other quadratic equation. If we let \(x = \cos{x}\), we can solve the equation: \( x^2 + 2x - 1 = 0 \). This equation can be solved using the quadratic formula: \( x = [-b ± sqrt(b^2 - 4ac)] / 2a \). From this we get: \( x = -1 ± sqrt(3) \). Therefore, the solutions for \( x \) are \( x = -1 + sqrt(3) \) or \( x = -1 - sqrt(3) \).

Step 4: Solve for angles

From the previous step, we identified that \( \cos{x} = -1 + sqrt(3) \) and \( \cos{x} = -1 - sqrt(3) \). However, both these values are outside the interval [-1,1] and thus, there are no solutions for x.

Key Concepts

Sine and Cosine Relationship

In trigonometry, sine and cosine are fundamental functions that describe the relationship between angles and the lengths of sides in a right triangle. They are often interchanged using trigonometric identities, which simplify complex trigonometric equations. One important identity that is crucial in solving equations like the one given in the exercise is the Pythagorean identity:

• \( \sin^2{x} + \cos^2{x} = 1 \)

This identity allows us to convert between sine and cosine easily. For example, if you have an equation involving \( \sin^2{x} \), you can substitute it with \( 1 - \cos^2{x} \). This is exactly what was done in Step 1 of the solution, and it allows us to transform the equation using only one trigonometric function. This simplification is often the first step in solving trigonometric equations as it reduces the complexity of the problem.
Understanding how to maneuver between these identities opens up easier paths to solve equations and is a foundational skill in higher-level math courses.

Quadratic Equations

After transforming the equation using the Pythagorean identity, we arrive at a form \( \cos^2{x} + 2\cos{x} - 1 = 0 \), which is a quadratic equation. A quadratic equation is a polynomial equation of the second degree, generally represented as:

• \( ax^2 + bx + c = 0 \)

Solving quadratic equations typically involves methods like factoring, completing the square, or using the quadratic formula:

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

In this exercise, the quadratic formula is used to find the potential solutions for \( \cos{x} \). The computed values, \( x = -1 + \sqrt{3} \) and \( x = -1 - \sqrt{3} \), show how algebraic techniques are essential to solve trigonometric equations. However, not all solutions derived from a quadratic equation will necessarily be viable in trigonometric contexts, as they must fall within certain boundaries like \([-1,1]\) for cosine values, which wasn't the case here.
By linking quadratic solutions back to trigonometric constraints, students can refine their answers and grasp how algebra and trigonometry intersect.

Pythagorean Identity

The Pythagorean identity, \( \sin^2{x} + \cos^2{x} = 1 \), not only links sine and cosine but also provides a tool for verification and simplification. It is grounded in the Pythagorean theorem from geometry, applied to the unit circle where the hypotenuse is always 1. This fundamental identity helps convert equations to obtain them in a form that uses either \( \sin \) or \( \cos \) for uniformity. By substituting \( \sin^2{x} \) with \( 1 - \cos^2{x} \), the problem morphs into a quadratic form that is easier to handle.
Students should remember that this identity is valid for all real numbers, ensuring versatility in applying it to a broad range of trigonometric problems. Testing the boundaries within which sine and cosine operate enriches understanding and problem-solving skills. Recognizing when and how to use the Pythagorean identity can simplify and unravel complex trigonometric equations.

Angles in Radians

In trigonometry, it is common to measure angles in radians instead of degrees. One complete revolution is \( 2\pi \) radians. This measure is preferred because it makes many mathematical formulas more elegant and simplifies the calculation, especially in calculus. When solving trigonometric equations, understanding radians is crucial for interpreting solutions correctly.

• An angle in radians ties directly to the unit circle, where \( \pi \) radians equal 180 degrees.
• An interval of \([0, 2\pi)\) translates to 0 to 360 degrees, excluding the endpoint \(2\pi\).

In the original exercise, we solve within the interval \([0, 2\pi)\), which means we're looking for angle solutions within one full circle. This further implies checking that any expression like \( \cos{x} \), derived from those solutions, must conform to the range of validated cosine values between -1 and 1.
Mastering the measurement of angles in radians aids in transitioning seamlessly between calculations and representations in different mathematical contexts.