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.)


sin 5

Verified step by step guidance

1. Recall that the sine function is positive in the first and second quadrants, and negative in the third and fourth quadrants of the unit circle.
2. Since the angle is given in radians, first identify which quadrant the angle 5 radians lies in by comparing it to multiples of \( \pi \) (approximately 3.14).
3. Calculate how many times \( \pi \) fits into 5 radians: \( \frac{5}{\pi} \approx 1.59 \), which means 5 radians is between \( \pi \) and \( 2\pi \).
4. Since \( \pi \) to \( 2\pi \) corresponds to the third and fourth quadrants, determine the exact quadrant by comparing 5 to \( \frac{3\pi}{2} \) (approximately 4.71).
5. Because 5 is greater than \( \frac{3\pi}{2} \), the angle lies in the fourth quadrant where sine values are negative, so \( \sin 5 \) is negative.

Key Concepts

Radian Measure and Unit Circle

Radian measure relates angles to the radius of a circle, where 2π radians equal 360°. Understanding the unit circle helps identify the position of an angle in standard position and its corresponding coordinates, which determine the signs of trigonometric functions.

Quadrantal Angles and Their Significance

Quadrantal angles are multiples of π/2 (90°) that lie on the x- or y-axis of the unit circle. Recognizing these angles helps in determining the sign of sine, cosine, and tangent functions in each quadrant without a calculator.

Sign of Sine Function in Different Quadrants

The sine function corresponds to the y-coordinate on the unit circle. It is positive in the first and second quadrants (0 to π radians) and negative in the third and fourth quadrants (π to 2π radians). This knowledge allows quick sign determination of sin(5).