How can I use the unit circle to find the inverse tan of 13 in degrees?

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To find the inverse tangent (or arctan) of a value using the concept of the unit circle, we first need to clarify the context of our question. The arctangent function returns the angle whose tangent is a given number. In this case, you are asking for the angle whose tangent is 13.

However, it’s important to note that the tangent function on the unit circle, defined as the ratio of the y-coordinate to the x-coordinate (tan(θ) = y/x), typically yields results for angles between -90° and 90° (or -π/2 and π/2 radians). The value of 13 is relatively high, meaning that the angle whose tangent is 13 would not be found directly on the unit circle since it exceeds the maximum value that tangent can reach for angles in this range.

Here’s the step-by-step approach to determine this:

  1. Understanding the Tangent Function: The tangent of an angle in a right triangle can be interpreted as the ratio of the side opposite the angle to the side adjacent to it. For large values (like 13), the opposite side is much longer than the adjacent side, representing very steep angles.
  2. Using the Inverse Function: To find the angle corresponding to tan(θ) = 13, you will use the inverse tangent function, denoted as tan-1(13).
  3. Calculating the Angle: You can use a scientific calculator or software that provides trigonometric functions. Make sure it is set to degrees if you want the answer in degrees. You’ll input:
  4. θ = tan-1(13)
  5. After calculation, you will find that θ is approximately 85.0° (depending on the precision of your tools).

Finally, since tangent is a periodic function with a period of 180° (or π radians), the general solution can be expressed as:

θ = 85° + n*180°

where n is any integer, although commonly we just consider the principal value within the range of -90° to 90° for the inverse function. Thus, the value of tan-1(13) in degrees is approximately 85°.

In summary, using the unit circle directly for tan(θ) = 13 isn’t possible due to the limits of what angles can be represented on the unit circle, but using the arctangent will give you an appropriate angle for the given tangent value.

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