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 ( ―1)

Verified step by step guidance

1. Identify the angle given: the problem asks about \( \sin(-1) \), where \(-1\) is in radians.
2. Recall that \( -1 \) radian is a negative angle, which means it is measured clockwise from the positive x-axis.
3. Determine the quadrant where the angle \( -1 \) radian lies. Since \( \pi \approx 3.14 \), and \( -1 \) is between \( 0 \) and \( -\pi/2 \) (which is approximately \(-1.57\)), the angle lies in the fourth quadrant.
4. Recall the sign of sine in each quadrant: sine is positive in the first and second quadrants, and negative in the third and fourth quadrants.
5. Since \( -1 \) radian is in the fourth quadrant, \( \sin(-1) \) is negative.

Key Concepts

Understanding Radian Measure

Radian measure is a way to express angles based on the radius of a circle. One full circle is 2π radians, so π radians equals 180 degrees. Knowing the approximate value of π (about 3.14) helps to locate angles on the unit circle and determine their quadrant.

Quadrantal Angles and Their Significance

Quadrantal angles are multiples of π/2 (90 degrees) that lie on the x- or y-axis of the unit circle. These angles divide the circle into four quadrants, each with specific signs for sine and cosine functions. Recognizing which quadrant an angle falls into helps determine the sign of trigonometric values.

Sign of the 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). Identifying the quadrant of the angle allows you to decide if sin(θ) is positive or negative without a calculator.