Ohio Assessments for Educators (OAE) Mathematics Practice Exam

How do you express the tangent of the sum of two angles, tan(α + β)?

1. (tanα + tanβ)/(1 - tanαtanβ)

2. (tanα + tanβ)/(1 + tanαtanβ)

3. (tanα - tanβ)/(1 - tanαtanβ)

4. (tanα - tanβ)/(1 + tanαtanβ)

The correct answer is: (tanα + tanβ)/(1 + tanαtanβ)

The tangent of the sum of two angles, tan(α + β), is expressed using the formula: \[ \tan(α + β) = \frac{\tan α + \tan β}{1 - \tan α \tan β} \] This formula arises from the properties of trigonometric functions and their relationships within the unit circle. When you add two angles, the tangent function combines the tangents of those angles in a specific manner to account for the changes in the direction of the angle (due to the sum). In this case, the correct answer indicates that the expression combines the tangents of the angles in the numerator (adding them together) and adjusts for the interaction between those angles in the denominator. Specifically, the subtraction in the denominator is key because it corrects for the elliptical nature of the tangent function when dealing with sums, preventing issues such as undefined values which can occur when the product of the tangents is equal to 1. The other options provided do not accurately reflect the correct interaction between the angles in the tangent function. For instance, the distinction between addition and subtraction in both necessitates the correct form which includes the negative sign in the denominator to ensure accuracy in calculating the sum of the angles.