Ohio Assessments for Educators (OAE) Mathematics Practice Exam

In the expression sin²θ + cos²θ = 1, what is the significance of this equation?

1. It describes the unit circle

2. It defines sine and cosine functions

3. It shows the relationship between angles

4. It represents the Pythagorean identity

The correct answer is: It represents the Pythagorean identity

The equation sin²θ + cos²θ = 1 is known as the Pythagorean identity in trigonometry. Its significance lies in its ability to express the fundamental relationship between the sine and cosine of an angle. This identity illustrates that for any angle θ, the sum of the squares of the sine and cosine functions equals one. This relationship is key in various applications of trigonometry, especially in the context of the unit circle. The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane, where any point on the circle can be represented in terms of coordinates (cos(θ), sin(θ)). The equation effectively confirms that the coordinates of any point on the unit circle satisfy this identity, reinforcing the importance of sine and cosine in circular motion and periodic functions. While the other options relate to important concepts in trigonometry, they do not capture the essence of the given equation as succinctly as identifying it as the Pythagorean identity. The choice that defines sine and cosine functions or describes the unit circle provides important context but mainly builds upon the foundational relationship articulated by the identity itself.