Solving Trigonometric Equations


For Exercises 51–54, solve for the angle θ, where 0 ≤ θ ≤ 2π.


sin² θ = cos² θ

Verified step by step guidance

1. Start by recognizing the given equation: sin² θ = cos² θ. This can be rewritten using the identity sin² θ + cos² θ = 1.
2. Rearrange the identity to express one trigonometric function in terms of the other: sin² θ = 1 - cos² θ.
3. Substitute this expression into the original equation: 1 - cos² θ = cos² θ.
4. Combine like terms to form a single equation: 1 = 2cos² θ.
5. Solve for cos² θ by dividing both sides by 2, then take the square root to find the possible values of cos θ. Consider the range 0 ≤ θ ≤ 2π to determine the specific angles that satisfy the equation.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. A key identity relevant to the given equation is the Pythagorean identity, which states that sin² θ + cos² θ = 1. Understanding these identities is crucial for simplifying and solving trigonometric equations.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angles that satisfy the equation within a specified interval. In this case, we need to manipulate the equation sin² θ = cos² θ to find the values of θ between 0 and 2π. Techniques often include using identities, factoring, and applying inverse trigonometric functions.

Unit Circle

The unit circle is a fundamental concept in trigonometry that provides a geometric interpretation of the sine and cosine functions. It is a circle with a radius of one centered at the origin of a coordinate plane. Understanding the unit circle helps in determining the angles corresponding to specific sine and cosine values, which is essential for solving the equation sin² θ = cos² θ.