Given the following unit circle, rotate green dot to the appropriate angle and then find the exact value of the function.tangent, 90, degreestan90 ∘

Break Down the Problem

1. Identify the position of 90 degrees on the unit circle.
2. Determine the value of the tangent function at that angle.

Relevant Concepts

• The tangent function $\tan(\theta)$ is defined as the ratio of the sine and cosine of the angle: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$
• The coordinates of points on the unit circle can be defined with respect to an angle $\theta$ as $(\cos(\theta), \sin(\theta))$.

Analysis and Detail

1. Position of 90 degrees: On the unit circle, the angle $90^\circ$ corresponds to the point $(0, 1)$ where:

   • $\cos(90^\circ) = 0$
   • $\sin(90^\circ) = 1$

2. Calculate $\tan(90^\circ)$: $\tan(90^\circ) = \frac{\sin(90^\circ)}{\cos(90^\circ)} = \frac{1}{0}$ Since division by zero is undefined, $\tan(90^\circ)$ does not yield a finite value.

Verify and Summarize

The calculation shows that $\tan(90^\circ)$ is undefined due to the cosine of 90 degrees being zero.

Final Answer

The exact value of $\tan(90^{\circ})$ is undefined.