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.)


cos 2

Verified step by step guidance

1. Identify the angle given in radians: 2 radians.
2. Recall that the cosine function is positive in the first and fourth quadrants and negative in the second and third quadrants.
3. Convert the angle to degrees if necessary, knowing that π radians is approximately 180 degrees, so 2 radians is approximately 2 * (180/π) degrees.
4. Determine the quadrant in which the angle 2 radians lies by comparing it to the quadrantal angles: 0, π/2, π, 3π/2, and 2π.
5. Conclude whether the cosine of the angle is positive or negative based on the quadrant it lies in.

Key Concepts

Quadrantal Angles

Quadrantal angles are angles that lie on the axes of the coordinate plane, specifically at 0, π/2, π, 3π/2, and 2π radians. These angles correspond to the points where the terminal side of the angle intersects the x-axis or y-axis. Understanding these angles is crucial for determining the sign of trigonometric functions, as they help identify which quadrant the angle lies in.

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 define the sine, cosine, and tangent of angles based on the coordinates of points on the circle. For any angle, the x-coordinate represents the cosine value, while the y-coordinate represents the sine value, which helps in determining their signs based on the angle's quadrant.

Sign of Trigonometric Functions

The sign of trigonometric functions (sine, cosine, tangent) varies depending on the quadrant in which the angle lies. In the first quadrant, all functions are positive; in the second, sine is positive; in the third, tangent is positive; and in the fourth, cosine is positive. Knowing the quadrant associated with a given angle is essential for determining whether the function values are positive or negative.