Without using a calculator, determine which of the two values is greater.
tan 1 or tan 2

Verified step by step guidance

1. Recall that the tangent function, \(\tan x\), is an increasing function on the interval \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), which means that if \(a < b\) and both are in this interval, then \(\tan a < \tan b\).
2. Identify the values given: \(1\) and \(2\) are in radians, and both lie within the interval \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) since \(\frac{\pi}{2} \approx 1.5708\) and \(2\) is slightly greater than \(\frac{\pi}{2}\), so we need to check the domain carefully.
3. Note that \(1\) radian is less than \(\frac{\pi}{2}\), but \(2\) radians is greater than \(\frac{\pi}{2}\), so \(\tan 2\) is not in the principal increasing interval and the tangent function has a vertical asymptote at \(x = \frac{\pi}{2}\).
4. Since \(\tan x\) approaches \(+\infty\) as \(x\) approaches \(\frac{\pi}{2}\) from the left and \(-\infty\) as \(x\) approaches \(\frac{\pi}{2}\) from the right, \(\tan 2\) (where \(2 > \frac{\pi}{2}\)) will be negative because it lies in the second interval where tangent is negative.
5. Therefore, compare the signs and values: \(\tan 1\) is positive (since \(1 < \frac{\pi}{2}\)) and \(\tan 2\) is negative (since \(2 > \frac{\pi}{2}\)), so \(\tan 1\) is greater than \(\tan 2\).

Key Concepts

Understanding the Tangent Function

The tangent function, tan(θ), is defined as the ratio of the sine and cosine of an angle θ (tan θ = sin θ / cos θ). It is periodic and increases on intervals where cosine is positive, with vertical asymptotes where cosine equals zero. Knowing its behavior helps compare values without a calculator.

Monotonicity of Tangent on Specific Intervals

On the interval (0, π/2), the tangent function is strictly increasing, meaning that if 0 < a < b < π/2, then tan(a) < tan(b). Since 1 and 2 are in radians and both lie within this interval, this property allows direct comparison of tan 1 and tan 2.

Radian Measure and Interval Placement

Angles measured in radians relate directly to the unit circle. Recognizing that 1 and 2 radians are between 0 and π/2 (~1.57) and π (~3.14) helps determine the behavior of trigonometric functions at these points. This context is essential for applying properties like monotonicity correctly.