Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles, and remember that π ≈ 3.14.)


tan 6.29

Verified step by step guidance

1. First, recognize that the angle given is in radians: 6.29 radians.
2. Recall that one full rotation around the unit circle is \(2\pi\) radians, which is approximately \(2 \times 3.14 = 6.28\) radians.
3. Since 6.29 radians is just a little more than \(2\pi\) radians, it means the angle has completed one full rotation and moved slightly beyond \(0\) radians (or \(0^\circ\)).
4. The tangent function has a period of \(\pi\) radians, so \(\tan(6.29)\) is equivalent to \(\tan(6.29 - 2\pi)\), which simplifies to \(\tan(6.29 - 6.28) = \tan(0.01)\) approximately.
5. Since \(\tan(0.01)\) is a very small positive angle in the first quadrant where tangent is positive, the value of \(\tan(6.29)\) is positive.

Key Concepts

Radian Measure and Quadrantal Angles

Radian measure expresses angles based on the radius of a circle, where 2π radians equal 360°. Quadrantal angles are multiples of π/2 (90°), such as 0, π/2, π, 3π/2, and 2π. Understanding where an angle lies relative to these helps determine the sign of trigonometric functions.

Periodic Nature of the Tangent Function

The tangent function has a period of π, meaning tan(θ) = tan(θ + nπ) for any integer n. This periodicity allows us to reduce large angles by subtracting multiples of π to find an equivalent angle within one period, simplifying the determination of the function's sign.

Sign of Tangent in Different Quadrants

Tangent is positive in the first and third quadrants and negative in the second and fourth. By locating the angle within the unit circle quadrants, one can decide the sign of tan(θ) without calculating its exact value.