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. Recognize that the problem involves determining the sign of \( \sin(-1) \) where the angle is in radians.
2. Recall that the sine function is positive in the first and second quadrants and negative in the third and fourth quadrants.
3. Convert the angle \(-1\) radian to its equivalent position on the unit circle. Since \(-1\) is negative, it means moving clockwise from the positive x-axis.
4. Note that \(-1\) radian is approximately \(-57.3\) degrees, which places it in the fourth quadrant.
5. Conclude that since the angle is in the fourth quadrant, \( \sin(-1) \) is negative.

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 and cosine functions take on specific values. Understanding these angles is crucial for determining the sign of trigonometric functions, as they help identify which quadrant the angle lies in.

Sine Function

The sine function, denoted as sin(θ), represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. Its values range from -1 to 1, and the sign of sin(θ) depends on the quadrant in which the angle θ is located. For quadrantal angles, the sine function takes on specific values: sin(0) = 0, sin(π/2) = 1, sin(π) = 0, and sin(3π/2) = -1.

Radian Measure

Radian measure is a way of measuring angles based on the radius of a circle. One radian is the angle formed when the arc length is equal to the radius of the circle. Understanding radian measures is essential for evaluating trigonometric functions without a calculator, as it allows for the identification of key angles and their corresponding sine and cosine values.