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. Convert the given radian measure to a more familiar form by considering its proximity to known angles. Since \( \pi \approx 3.14 \), we know that \( 2\pi \approx 6.28 \).
2. Recognize that \( 6.29 \) radians is slightly more than \( 2\pi \) radians, which means it is just past a full circle (360 degrees).
3. Since \( 6.29 \) radians is just past \( 2\pi \), it is in the first quadrant of the unit circle.
4. Recall that in the first quadrant, the tangent function is positive because both sine and cosine are positive.
5. Conclude that \( \tan(6.29) \) is positive because it is in the first quadrant.

Key Concepts

Quadrantal Angles

Quadrantal angles are angles that lie on the axes of the Cartesian coordinate system, specifically at 0, π/2, π, 3π/2, and 2π radians. These angles correspond to the points where the sine or cosine functions take on specific values, either 0, 1, or -1. Understanding these angles is crucial for determining the sign of trigonometric functions in different quadrants.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of the coordinate plane. It is a fundamental tool in trigonometry, as it allows us to visualize the values of sine, cosine, and tangent for various angles. By knowing the position of an angle on the unit circle, we can easily determine the signs of these trigonometric functions based on the quadrant in which the angle lies.

Tangent Function

The tangent function, defined as the ratio of the sine to the cosine of an angle (tan(θ) = sin(θ)/cos(θ)), indicates the slope of the line formed by the angle in the unit circle. The sign of the tangent function depends on the signs of sine and cosine in the respective quadrant. In the first and third quadrants, tangent is positive, while in the second and fourth quadrants, it is negative.