Concept Check Work each problem. Without using a calculator, determine which of the following numbers is closest to sin 115°: -0.9, -0.1, 0, 0.1, or 0.9.

Verified step by step guidance

1. Recall that the sine function is positive in the second quadrant, where angles range from 90° to 180°. Since 115° is in the second quadrant, \(\sin 115^\circ\) is positive.
2. Use the reference angle to find the sine value. The reference angle for 115° is \(180^\circ - 115^\circ = 65^\circ\). So, \(\sin 115^\circ = \sin 65^\circ\).
3. Remember that \(\sin 65^\circ\) is a positive value less than 1 but closer to 1 than to 0, because 65° is closer to 90° than to 0°.
4. Compare the given options (-0.9, -0.1, 0, 0.1, 0.9) with the expected value of \(\sin 65^\circ\). Since \(\sin 65^\circ\) is positive and relatively large, the closest number should be positive and near 0.9.
5. Conclude that among the given options, 0.9 is the closest to \(\sin 115^\circ\) based on the properties of the sine function and the reference angle.

Key Concepts

Unit Circle and Angle Measurement

The unit circle represents angles and their corresponding sine values on a circle of radius one. Angles are measured in degrees or radians, and sine corresponds to the y-coordinate of the point on the circle. Understanding where 115° lies helps estimate the sine value without a calculator.

Sine Function Properties in Different Quadrants

Sine values vary depending on the quadrant of the angle. Since 115° is in the second quadrant (90° to 180°), sine values are positive and decrease from 1 at 90° to 0 at 180°. This knowledge helps narrow down the possible sine values for 115°.

Reference Angles and Approximation

A reference angle is the acute angle formed with the x-axis, used to find sine values of angles in other quadrants. For 115°, the reference angle is 65°. Knowing sine 65° is about 0.91 helps approximate sine 115° as positive and close to 0.9.