91Ó°ÊÓ

The cosine function is one of the fundamental components of trigonometry, which describes the ratio of the adjacent side to the hypotenuse of a right-angled triangle. It is critical in understanding the properties of circles and waves and has a deep connection with the unit circle. For any angle \( \theta \), the cosine can be found by measuring the x-coordinate of the point on the unit circle where the terminal side of \( \theta \) intersects it.

In the context of the exercise, the cosine of the angle \( \frac{3 \pi}{4} \) is considered. Since \( \frac{3 \pi}{4} \) lies in the second quadrant of the unit circle, where the x-coordinate is negative, the cosine value is negative as well. This result underscores the importance of understanding the sign of trigonometric functions in different quadrants, which is critical when applying these concepts to various problems.

Radians provide a way to express angles based on the arc length and the radius of a circle. One radian is the angle created when the arc length is exactly equal to the radius of the circle. The significance of using radian measure lies in its natural fit with circular motion and trigonometric functions

The exercise involves \( \frac{3 \pi}{4} \) radians, which can be visualized on the unit circle. The unit circle, by definition, has a radius of one, making the radian measure particularly convenient. When converting degrees to radians, the factor \( \frac{\pi}{180} \) is used, reinforcing the connection between circular geometry and these measures. It is essential to become comfortable with radians to adeptly navigate trigonometry.

Significant digits (also called significant figures) are a form of precision representation in numerical answers. They indicate the number of meaningful digits in a number, starting with the first non-zero digit. Proper use of significant digits is crucial in scientific calculations to avoid overrepresenting the precision of a measurement or computed value.

In our exercise, the solution is to be evaluated to four significant digits. However, the squared cosine value \( \frac{1}{2} \) has only one significant digit. When a result has fewer significant digits than required, as in this case, it is often presented with trailing zeros (up to the needed significant digits) or in scientific notation to convey the correct level of precision. For example, \( \frac{1}{2} \) could be written as 0.5000 to reflect four significant digits.