Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the relationship between sine, cosine, and tangent?

1. tan(x) = sin(x) * cos(x)

2. tan(x) = sin(x) - cos(x)

3. tan(x) = sin(x) / cos(x)

4. tan(x) = cos(x) - sin(x)

The correct answer is: tan(x) = sin(x) / cos(x)

The relationship between sine, cosine, and tangent in trigonometry is defined in terms of a right triangle. Tangent is specifically the ratio of the length of the opposite side to the length of the adjacent side. In terms of sine and cosine, this relationship is articulated as the tangent of an angle being equal to the sine of that angle divided by the cosine of that angle. Mathematically, this is expressed as: tan(x) = sin(x) / cos(x) This definition arises directly from the definitions of the sine and cosine functions. The sine function relates to the ratio of the side opposite the angle to the hypotenuse, while the cosine function relates to the ratio of the adjacent side to the hypotenuse. By dividing these two ratios, you derive the tangent function. The other options do not reflect the fundamental relationship applicable within the unit circle or any right triangle context, which further supports the correctness of the ratio provided in the correct choice. Using the established definitions of sine, cosine, and tangent allows one to correctly identify the mathematical relationships and derive properties or solve equations involving these functions.