Tan-1-1 Without Calculator

What is tan-1-1?

The notation tan-1-1 typically refers to the arctangent of -1, which is the inverse trigonometric function of the tangent. The arctangent function, denoted as tan-1(x) or arctan(x), returns the angle whose tangent is x. For x = -1, we're looking for the angle θ such that tan(θ) = -1.

The result of tan-1(-1) is an angle in the range of -π/2 to π/2 radians (or -90° to 90°). The exact value is π/4 radians (45°) because tan(π/4) = -1.

Methods to calculate tan-1-1

There are several approaches to calculate tan-1-1 without a calculator:

1. Unit circle method: Using the properties of the unit circle to find the angle where the tangent is -1.
2. Geometric interpretation: Constructing a right triangle where the opposite side is -1 and the adjacent side is 1.
3. Series expansion: Using the Taylor series expansion of the arctangent function.

The unit circle method is the most straightforward for this calculation.

Step-by-step calculation

Here's how to calculate tan-1-1 using the unit circle method:

1. Recall that tan(θ) = opposite/adjacent in a right triangle.
2. Set up the equation tan(θ) = -1, which implies opposite/adjacent = -1.
3. This can be satisfied by a right triangle with opposite side length -1 and adjacent side length 1.
4. In the unit circle, this corresponds to the angle π/4 radians (45°) in the fourth quadrant where the tangent is negative.
5. Therefore, tan-1(-1) = π/4 radians.

Verification

To verify that tan-1-1 = π/4 radians, we can check:

• tan(π/4) = sin(π/4)/cos(π/4) = (√2/2)/(√2/2) = 1
• However, in the fourth quadrant, both sine and cosine are positive, so tan(π/4) is actually 1.
• This suggests a correction is needed in the interpretation.

The correct interpretation is that tan-1(-1) = -π/4 radians because tan(-π/4) = -1. The angle -π/4 is equivalent to 3π/4 radians in the second quadrant where tangent is negative.