Problem 71 - Algebra and Trigonometry

Question

Solve: \(2 \cos ^{2} x+3 \sin x-3=0, \quad 0 \leq x<2 \pi\)

Step-by-Step Solution

Answer

The solution to the given equation \(2 \cos ^{2} x+3 \sin x-3=0, \quad 0 \leq x<2 \pi\) is \(x = 0\).

Step 1: Rewriting the equation

Rewrite the equation using the Pythagorean identity \( \sin ^{2} x = 1 - \cos ^{2} x \), this will helps to simplify the equation in terms of one trigonometric function. Hence, the given equation \(2 \cos ^{2} x+3 \sin x-3=0\) can be rewritten as \(2 \cos^{2} x + 3(1 - \cos ^{2} x) - 3 = 0\).

Step 2: Simplify the equation

Further simplifying gives \( -\cos ^{2} x + 3 \cos x - 3 = 0 \), which can be represented in quadratic form of \( ax^{2} + bx + c = 0 \) where \( a = -1, b =3, c = -3 \) with \( x \) replaced by \( \cos x \) here.

Step 3: Solve the quadratic equation

Apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) which gives \( \cos x = \frac{-3 \pm \sqrt{9 - 4*(-1)*(-3)}}{2*(-1)} = 1, 3 \). Note that \(\cos x = 3\) is not possible as the cosine values lie between -1 and 1.

Step 4: Find \(x\) values

Since the valid solution is \(\cos x = 1\), find the \(x\) values in the range [0, 2π), that satisfy the equation. The cos function equals 1 at \(x = 0\) and \(x = 2\pi\). Since our interval is closed at 0 and open at \(2\pi\), the unique solution in the interval [0, \(2\pi\)) is \(x = 0\).

Key Concepts

Pythagorean Identity

Understanding the Pythagorean identity is key in solving many trigonometric equations, as it allows us to express one trigonometric function in terms of another. The Pythagorean identity is a fundamental relation in trigonometry, represented by the equation \( \sin^2 x + \cos^2 x = 1 \).

This identity derives from the Pythagorean theorem, which refers to the relationship between the sides of a right-angled triangle. In trigonometric functions, the identity is used to simplify problems by converting between \( \sin x \) and \( \cos x \) or vice versa. In the context of our exercise, it was cleverly employed to translate a combined sine and cosine equation into a form involving only cosine, making it manageable as a quadratic equation. This is an essential technique for solving trigonometric equations efficiently.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree that have the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. A variety of methods exist for solving quadratic equations, such as factoring, completing the square, and using the quadratic formula.

The quadratic formula, \( x = \frac{{-b \pm \sqrt{{b^{2} - 4ac}}}}{{2a}} \), is particularly useful for finding the solutions to any quadratic equation when factoring is difficult or not possible. In our exercise, after using the Pythagorean identity to simplify the trigonometric equation to a quadratic form, we indeed employed the quadratic formula to solve for \( \cos x \) values. This not only presented the solution in an accessible form but also illuminated which answers were viable given the range of the cosine function.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides, which are extremely useful in various areas such as geometry, wave motion, and periodic phenomena. The most common trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), each with specific ranges and properties.

For the cosine function, the range is between -1 and 1, inclusive. This aspect was crucial in the given exercise, where an understanding of the bounds of \( \cos x \) helped to eliminate \( \cos x = 3 \) as an extraneous solution. Knowing how trigonometric functions behave and their values at specific angles not only helps in solving equations but also ensures the correctness of the solutions within a given interval, as demonstrated in the final step of the exercise where the solution \( \cos x = 1 \) corresponded to \( x = 0 \) within the interval \( [0, 2\pi) \).