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. Convert the angle from radians to degrees: Since \( \pi \approx 3.14 \), we have \( 5 \approx \frac{5}{3.14} \times 180 \) degrees.
2. Determine the equivalent angle within the standard 0 to 360-degree range by subtracting multiples of 360 degrees if necessary.
3. Identify the quadrant in which the angle lies: Recall that angles between 0° and 90° are in the first quadrant, 90° to 180° in the second, 180° to 270° in the third, and 270° to 360° in the fourth.
4. Recall the sign of the sine function in each quadrant: Sine is positive in the first and second quadrants, and negative in the third and fourth quadrants.
5. Based on the quadrant identified, determine whether \( \sin 5 \) is positive or 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: sin(0) = 0, sin(π/2) = 1, sin(π) = 0, sin(3π/2) = -1, and sin(2π) = 0. Understanding these angles is crucial for determining the sign of trigonometric functions.

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 visualize the values of sine and cosine for any angle. The coordinates of points on the unit circle correspond to the cosine and sine of the angle, respectively, helping to determine whether these values are positive or negative based on the angle's location in the coordinate system.

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. In this context, understanding how to convert degrees to radians and recognizing common radian values (like π/6, π/4, π/3, etc.) is essential for evaluating trigonometric functions without a calculator, especially when determining the sign of sine at specific angles.