Problem 68 - Precalculus Essentials

Question

Use an identity to solve each equation on the interval \([0,2 \pi)\) $$3 \cos ^{2} x=\sin ^{2} x$$

Step-by-Step Solution

Answer

The given equation \(3 \cos ^{2} x=\sin ^{2} x\) has no solutions within the interval \([0,2 \pi)\).

Step 1: Substitute \( \sin ^2(x)\) with \(1 - \cos ^2(x)\)

The equation can be written as \(3 \cos ^{2} x= 1 - \cos ^2(x)\).

Step 2: Rearrange into standard quadratic form

To solve for \( \cos(x)\), rearrange the equation into a standard quadratic form: \(4 \cos ^{2} x + 3 = 0\).

Step 3: Solve the quadratic equation to get solutions for \( \cos(x)\)

Solve the quadratic equation to get \( \cos ^{2} x = -3/4\). However, since the square of cosine function always lies between 0 and 1, there is no real-valued solution within the range.

Step 4: Check the solution within the interval \([0,2 \pi)\)

Since there are no valid solutions for \( \cos(x)\) there also can be no valid solutions for \( x\) within the interval \([0,2 \pi)\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in mathematics. They follow the standard form of \(ax^2 + bx + c = 0\). In these equations, "\(a\)", "\(b\)", and "\(c\)" are constants, where "\(a\)" is not zero. This type of equation forms a parabola when graphed, and its solutions can be found using various methods. The main techniques include:

• Factoring: Finding two numbers that multiply to "\(ac\)" and add to "\(b\)".
• The Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the Square: Rewriting the equation in the form \((x - p)^2 = q\).

When working with quadratic equations, it is crucial to identify relevant values by setting the equation to zero. This relates directly to solving trigonometric equations, where we often transform the equation into a quadratic form. In our example problem, replacing \(\sin^2(x)\) with \(1 - \cos^2(x)\) led to forming a quadratic equation in terms of \(\cos^2(x)\). This transformation is essential for analyzing and solving the equations accurately.

Trigonometric Functions

Trigonometric functions are fundamental to mathematics, arising mainly from right-angled triangles and circles. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each of these functions represents a ratio of sides in a right-angled triangle. Cosine, for instance, is the ratio of the adjacent side to the hypotenuse.

• Sine (\(\sin\)) function: Opposite / Hypotenuse
• Cosine (\(\cos\)) function: Adjacent / Hypotenuse
• Tangent (\(\tan\)) function: Opposite / Adjacent

In this context, these functions describe the periodic behavior of waves, providing efficient models for oscillating phenomena. Each function has well-established identities that simplify and facilitate the solving process of complex equations, like transforming \(\sin^2(x)\) using the identity \(\sin^2(x) = 1 - \cos^2(x)\). Understanding these identities and how they relate is crucial for effectively rearranging and solving equations in trigonometry.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the variable (often \(x\)) that satisfy an identity or equation. These equations often require manipulating trigonometric identities to simplify and solve. When solving the problem \(3 \cos^2(x) = \sin^2(x)\), we utilized the identity \(\sin^2(x) = 1 - \cos^2(x)\) to express both sides in terms of \(\cos^2(x)\). This allowed us to reformulate the trigonometric equation into a quadratic equation: \(4 \cos^2(x) - 1 = 0\). Once simplified, the next step involves solving the quadratic equation. However, during the solving process, it was revealed that there were no valid real solutions (as \(\cos^2(x)\) cannot be negative), indicating no solutions to the trigonometric equation within the given interval \([0, 2\pi)\). Thoroughly checking the solutions within the specified domain is vital. This ensures we find all possible solutions accurately and confirms whether the equation holds true under given conditions, a critical step in solving trigonometric equations.