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How is tan 1 calculated?

Published in Trigonometry Calculations 2 mins read

The question "How is tan 1 calculated?" can be interpreted in a few ways, and each interpretation leads to a different answer. We will address two common interpretations here:

  1. How to calculate the tangent of 1 radian?
  2. How to calculate the angle whose tangent is 1? (This is related to tan-1(1)).

Calculating tan(1 radian)

The tangent of 1 radian (approximately 57.3 degrees) is not something you would typically calculate by hand. Here's how it's generally determined:

  • Using a calculator: Most scientific calculators have a tan function. Simply input 1 (make sure your calculator is in radian mode), and it will return the value.
  • Using mathematical software: Software like MATLAB, Python with NumPy, and similar tools can readily calculate tan(1).
  • Approximation methods: There are infinite series expansions, such as the Taylor series, that can be used to approximate the tangent function. These methods are more involved and require a good understanding of calculus.

The result is approximately tan(1 radian) ≈ 1.5574.

Calculating tan-1(1)

The question can be interpreted as finding the angle (in radians or degrees) whose tangent is equal to 1. This is often written as tan-1(1) or arctan(1).

Understanding Inverse Tangent

The reference tells us that the inverse tangent, denoted by tan-1, is used to find the angle θ when we know the ratio of the opposite side to the adjacent side in a right triangle. The inverse tangent formula is given by:

θ = tan-1[(opposite side) / (adjacent side)]

Calculating tan-1(1)

If tan(θ) = 1, then using the inverse tangent, we have θ = tan-1(1).

  • Identifying the angle: We know that in a right-angled triangle, the tangent of an angle is the ratio of its opposite side to the adjacent side. If the tangent is 1, the opposite and adjacent sides must be equal. This happens when the angle is 45 degrees or π/4 radians.

  • Result:

    • In degrees: tan-1(1) = 45°
    • In radians: tan-1(1) = π/4 radians (approximately 0.7854)

Summary

Calculation Value Units Method
tan(1) Approximately 1.5574 Radians Calculator or mathematical software, or using a series expansion (advanced).
tan-1(1) 45 degrees or π/4 radians Degrees/Radians Understanding the relationship between angles and ratios in a right triangle or calculator.

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