Problem 4 - Algebra and Trigonometry

Question

\(3-16 \cdot\) Solve the given equation. $$ \sin ^{2} \theta=4-2 \cos ^{2} \theta $$

Step-by-Step Solution

Answer

No real solution exists for \( \theta \).

Step 1: Recognize Trigonometric Identity

Recall the Pythagorean identity for sine and cosine: \( \sin^2 \theta + \cos^2 \theta = 1 \). Use this identity to express \( \sin^2 \theta \) in terms of \( \cos^2 \theta \).

Step 2: Substitute in the Identity

Substitute \( \sin^2 \theta = 1 - \cos^2 \theta \) into the given equation, resulting in \( 1 - \cos^2 \theta = 4 - 2 \cos^2 \theta \).

Step 3: Simplify the Equation

Rearrange the equation to isolate terms involving \( \cos^2 \theta \) on one side: \( 1 + \cos^2 \theta = 4 \).

Step 4: Solve for \( \cos^2 \theta \)

Subtract 1 from both sides to get \( \cos^2 \theta = 3 \).

Step 5: Analyze the Result

As \( \cos^2 \theta = 3 \) is not possible (since \( \cos^2 \theta \) ranges from 0 to 1 for real numbers), there are no real solutions for \( \theta \).

Key Concepts

Trigonometric Identity

Trigonometric identities are equations that hold true for all values of the variables where both sides of the equation are defined. These identities are crucial tools in solving trigonometric equations as they allow us to manipulate and simplify expressions.
In trigonometry, several identities can help express one trigonometric function in terms of others. One of the most important identities is the Pythagorean identity. By recognizing and utilizing these identities, it becomes easier to solve complex equations.
For instance, if we have an equation involving \( \sin^2 \theta \) and \( \cos^2 \theta \), knowing how to use a trigonometric identity like the Pythagorean identity can simplify the process of finding the solution.

Pythagorean Identity

The Pythagorean identity is perhaps the most iconic identity in trigonometry. It relates the squares of the sine and cosine functions to the unit circle. The identity is given by:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is derived from the properties of a right-angled triangle and the definition of the sine and cosine functions on the unit circle.
In the given exercise, the use of the Pythagorean identity was key in expressing \( \sin^2 \theta \) in terms of \( \cos^2 \theta \). Specifically, the identity allowed us to replace \( \sin^2 \theta \) with \( 1 - \cos^2 \theta \) in the equation \( \sin^2 \theta = 4 - 2 \cos^2 \theta \). This substitution is a strategic step that simplifies the problem significantly, making the solution more accessible.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that are commonly encountered in mathematics, especially when dealing with angles and right triangles.
These functions have several essential properties. For instance:

• The range of \( \sin \theta \) and \( \cos \theta \) is between -1 and 1.
• \( \sin^2 \theta + \cos^2 \theta = 1 \) as mentioned in the Pythagorean identity.

In the exercise, the equation \( \sin^2 \theta = 4 - 2 \cos^2 \theta \) involved both \( \sin \) and \( \cos \). The transformation of \( \sin^2 \theta \) to \( 1 - \cos^2 \theta \) used the relationship between these two functions to simplify and solve the equation.
Understanding the relationship between sine and cosine allows us to use substitutions that reveal solutions or, in this case, show that a solution does not exist within the real number system.

Real Number Solutions

Real number solutions refer to solutions of an equation that are part of the set of real numbers, which include all rational and irrational numbers. In trigonometric problems, finding real solutions is common when working within the domain of real numbers, which is essentially the span of all possible angles for sine and cosine functions.
In this specific exercise, after the substitution using the Pythagorean identity and simplification, we reached the equation \( \cos^2 \theta = 3 \). Since \( \cos^2 \theta \) can only be within the range of 0 to 1 for real angles, the result of \( \cos^2 \theta = 3 \) indicates that there are no real number solutions.
Thus, understanding the constraints of sine and cosine values within the context of real numbers is crucial when analyzing the viability of potential solutions.