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, which is 2 radians in this case.
2. Recall that \( \pi \approx 3.14 \), so 2 radians is less than \( \pi \) but greater than \( \frac{\pi}{2} \) (approximately 1.57). This means the angle lies in the second quadrant of the unit circle.
3. Remember the signs of cosine in each quadrant: cosine is positive in the first and fourth quadrants, and negative in the second and third quadrants.
4. Since 2 radians is in the second quadrant, where cosine values are negative, conclude that \( \cos 2 \) is negative.
5. Thus, without calculating the exact value, you can determine the sign of \( \cos 2 \) by understanding the position of the angle on the unit circle.

Key Concepts

Understanding Radian Measure and Quadrantal Angles

Radian measure relates angles to the radius of a circle, where π radians equal 180°. Quadrantal angles are multiples of π/2 (90°), dividing the unit circle into four quadrants. Knowing where an angle lies helps determine the sign of trigonometric functions without a calculator.

Unit Circle and Sign of Trigonometric Functions

The unit circle defines sine and cosine values based on coordinates of points on the circle. Cosine corresponds to the x-coordinate, which is positive in the first and fourth quadrants and negative in the second and third. Identifying the quadrant of the angle 2 radians helps decide the sign of cos 2.

Approximation of π and Angle Location

Knowing π ≈ 3.14 allows estimation of where 2 radians lies on the unit circle. Since 2 is less than π (3.14) but greater than π/2 (1.57), the angle is in the second quadrant. This approximation is crucial to determine the sign of cosine without exact calculation.